1. Construction for Physics-informed Neural Networks (PINN)¶
- Neural Networks play the role as modeling to approximate the PDE solution with given condition.
- The Given conditions(Data-driven) solutions in PDE / ODE are set as follows:
where $u(t,x)$ denotes the solution, $N[.; \lambda]$ is a nonlinear operator with parameter $\lambda$, and the domain $\Omega$ is in $\mathbb{R}^D$.
2. Solving PDE/ ODE from Neural Networks¶
1. PDE / ODE given boundary condition and initial condition setting¶
The concept involves employing neural networks to solve differential equations. This is achieved by utilizing a neural network to represent the solution and training it to meet the conditions specified by initial condition and boundary condition of the differential equation.
Consider a system of partial differential equations:
$ \frac{du}{dt} = f(u, t) $ where $ t$ is in $[0, 1]$ and
$ u(x, 0) = u_0 $ as given boundary condition.
$ u(0, t) = g_1(x)$ and $ u(L, t) = g_2(x)$ as given initial condition where $\Omega = [0, L]$.
2. Solving the PDE / ODE¶
To solve this, we approximate the solution as a result of a neural network:
$$ NN(t) \approx u(t) $$For the assumption as $NN(t)$ were the true solution, then we know $ NN'(t) = f(NN(t), t) $.
3. Error estimaties by Loss(Cost) function¶
Based on the above $NN(t)$, our loss function is:
$$L(\omega) = \sum_i \left(\frac{dNN(t_i)}{dt} - f(NN(t_i), t_i)\right)^2 $$The $ t_i $ can be chosen in various ways, such as randomly or on a grid. When this loss function is minimized, we have $ \frac{dNN(t_i)}{dt} \approx f(NN(t_i), t_i) $, and thus $ NN(t)$ approximately solves the differential equation.
On the PDE domain $ \Omega \times [0, t_0]$ of $(x,t)$, to reflect the boundary condition $u_0 = u (x, 0)$ and boundary condition $u(0, t), u(L, t)$ of our solution, these terms is added to the loss function:
$$ L(\omega) = \sum_i [ \left(\frac{dNN(t_i)}{dt} - f(NN(t_i), t_i)\right)^2 + (NN(0) - u_0)^2 + (NN(0,t) - u(0,t))^2+ (NN(L,t) - u(L,t))^2] $$Here, $ \omega $ represents the parameters defining the neural network $ NN $ that approximates $ u $. This process result in finding weights that minimize the loss function.
3. Example - One Dimensional Heat Equation¶
In [38]:
import torch
import torch.nn as nn
from torch.autograd import Variable
import numpy as np
In [39]:
device = torch.device("cpu")
3.1 Define the class for Neural Network¶
We designate Net as our solution $u_\omega (x, t)$.
When constructing the network, it's crucial to consider the number of inputs and outputs. For the input variables, there are x and t. For the output variables, there is one variable u.
We can incorporate as many hidden layers as needed, each with as many neurons as desired. A more intricate network is better equipped to discover complex solutions, but it also demands more data.
Now, create this network as the following codes. Minimum 8 hidden layers, each containing 5 neurons. The first def code indicates linear transformation of Neural Network, and the second def code means transfromation through Sigmoid activation function.
In [40]:
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.hidden_layer1 = nn.Linear(2,100) # number of input variables = 2, output variables = 5
self.hidden_layer2 = nn.Linear(100,100)
self.hidden_layer3 = nn.Linear(100,100)
self.hidden_layer4 = nn.Linear(100,100)
self.hidden_layer5 = nn.Linear(100,100)
self.hidden_layer6 = nn.Linear(100,100)
self.hidden_layer7 = nn.Linear(100,100)
self.hidden_layer8 = nn.Linear(100,100)
self.output_layer = nn.Linear(100,1)
def forward(self, x,t):
inputs = torch.cat([x,t],axis=1) # Merged two single-column arrays into a single array with two columns.
layer1_out = torch.sigmoid(self.hidden_layer1(inputs)) #Sigmoid function -> σ(x)= 1/(1+exp(-x))
layer2_out = torch.sigmoid(self.hidden_layer2(layer1_out))
layer3_out = torch.sigmoid(self.hidden_layer3(layer2_out))
layer4_out = torch.sigmoid(self.hidden_layer4(layer3_out))
layer5_out = torch.sigmoid(self.hidden_layer5(layer4_out))
layer6_out = torch.sigmoid(self.hidden_layer6(layer5_out))
layer7_out = torch.sigmoid(self.hidden_layer7(layer6_out))
layer8_out = torch.sigmoid(self.hidden_layer8(layer7_out))
output = self.output_layer(layer8_out)
## For regression, no activation is used in output layer
return output
# Result of the above class:
# transformation x,t -> u -> sigmoid(u) through each hiddeon layer
3.2 Weight update setting - Optimizer¶
To iterate the nueral network, we update the weight of the linear model we set in chapter 2 by using Adam Algorithm.
In [41]:
from torch.optim import Adam
net = Net()
net = net.to(device)
# Set cost function as Mean squared error
mse_cost_function = torch.nn.MSELoss()
#Run Adam algorithm with output of the neural network processing.
optimizer = torch.optim.Adam(net.parameters())
3.3 Setting PDE with drawn u(x,t)¶
Employing the u(x,t) in the previous step, we set the 1-dimensional equation function f(x,t) for $ x \in [-2, 2]$ and $ t \in [0,1]$. Also we set its loss(cost) function to optimize the error bound. The Heat equation setting is as follows with the code below:
The function $u(x, t)$, representing heat flow, needs to adhere to the partial differential equation below. $$u_t = u_{xx}$$ Moreover, we impose two additional conditions. Initially, we set the temperature at both ends of the domain of $x$, specifying u(0, t) and u(L, t). Therefore, we assume that both ends are maintained at a temperature of 0 degrees Celsius:
In [42]:
# Partial Differential Equation (PDE) and its associated loss function.
def pde_loss(x, t, net):
# The dependent variable 'u' is obtained from the network based on independent variables x and t.
u = net(x, t)
# Before defining the PDE = du/dt - d^2(u)/(dx)^2, we need to compute du/dx and du/dt.
u_x = torch.autograd.grad(u,x,torch.ones(u.shape),retain_graph=True,create_graph=True)[0]
#torch.autograd.grad(u.sum(), x, create_graph=True)[0]
u_xx = torch.autograd.grad(u_x,x,torch.ones(u_x.shape),create_graph=True)[0]
#torch.autograd.grad(u_x.sum(), x, create_graph=True)[0]
# torch.autograd.grad <- Computes and returns the sum of gradients of outputs with respect to the inputs.
u_t = torch.autograd.grad(u.sum(), t, create_graph=True)[0]
pde = u_t - u_xx
return pde
3.4 Boundary condition setting¶
Boundary condition for the heat equation is given as $u(x,0) = sin(\pi x)$ as follows. For the training data on the domain, is it used as the initial data.
In [43]:
# Data for Boundary Conditions
# Initial condition: u(x,0) = sin(pi * x)
# Boundary conditions provide initial data points for training.
# Boundary conditions state that for any x in the range [-2, 2] and time in [0, 1], the value of u is given by sin(pi * x).
# Generate 500 random numbers for x
x_bc = (2 - (-2)) *torch.rand(500,1) - 2
#np.random.uniform(low=0.0, high=6.0, size=(500,1))
t_bc = torch.zeros(500,1)
t_inner = torch.rand(500,1)
#np.zeros((500,1))
# Compute u based on the boundary condition: u(x,0) = sin(pi * x)
u_initial1 = t_bc
u_initial2 = t_bc
u_bc = np.sin(np.pi *x_bc)
3.5.1 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.1]$¶
- training with iteration 20000
We train the data while optimizing the weight in the linear model of neural network.
In [44]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.1, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.6134) 1 Training Loss: tensor(0.5461) 2 Training Loss: tensor(0.5072) 3 Training Loss: tensor(0.4940) 4 Training Loss: tensor(0.4995) 5 Training Loss: tensor(0.5122) 6 Training Loss: tensor(0.5215) 7 Training Loss: tensor(0.5231) 8 Training Loss: tensor(0.5183) 9 Training Loss: tensor(0.5102) 10 Training Loss: tensor(0.5022) 11 Training Loss: tensor(0.4964) 12 Training Loss: tensor(0.4940) 13 Training Loss: tensor(0.4946) 14 Training Loss: tensor(0.4972) 15 Training Loss: tensor(0.5002) 16 Training Loss: tensor(0.5024) 17 Training Loss: tensor(0.5031) 18 Training Loss: tensor(0.5021) 19 Training Loss: tensor(0.5000) 20 Training Loss: tensor(0.4975) 21 Training Loss: tensor(0.4954) 22 Training Loss: tensor(0.4941) 23 Training Loss: tensor(0.4939) 24 Training Loss: tensor(0.4945) 25 Training Loss: tensor(0.4955) 26 Training Loss: tensor(0.4964) 27 Training Loss: tensor(0.4970) 28 Training Loss: tensor(0.4969) 29 Training Loss: tensor(0.4963) 30 Training Loss: 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Training Loss: tensor(0.0055) 19825 Training Loss: tensor(0.0053) 19826 Training Loss: tensor(0.0049) 19827 Training Loss: tensor(0.0036) 19828 Training Loss: tensor(0.0029) 19829 Training Loss: tensor(0.0037) 19830 Training Loss: tensor(0.0030) 19831 Training Loss: tensor(0.0022) 19832 Training Loss: tensor(0.0024) 19833 Training Loss: tensor(0.0025) 19834 Training Loss: tensor(0.0032) 19835 Training Loss: tensor(0.0044) 19836 Training Loss: tensor(0.0060) 19837 Training Loss: tensor(0.0074) 19838 Training Loss: tensor(0.0094) 19839 Training Loss: tensor(0.0111) 19840 Training Loss: tensor(0.0124) 19841 Training Loss: tensor(0.0143) 19842 Training Loss: tensor(0.0161) 19843 Training Loss: tensor(0.0187) 19844 Training Loss: tensor(0.0220) 19845 Training Loss: tensor(0.0257) 19846 Training Loss: tensor(0.0289) 19847 Training Loss: tensor(0.0251) 19848 Training Loss: tensor(0.0144) 19849 Training Loss: tensor(0.0057) 19850 Training Loss: tensor(0.0050) 19851 Training Loss: tensor(0.0091) 19852 Training Loss: tensor(0.0139) 19853 Training Loss: tensor(0.0146) 19854 Training Loss: tensor(0.0065) 19855 Training Loss: tensor(0.0047) 19856 Training Loss: tensor(0.0089) 19857 Training Loss: tensor(0.0080) 19858 Training Loss: tensor(0.0083) 19859 Training Loss: tensor(0.0047) 19860 Training Loss: tensor(0.0042) 19861 Training Loss: tensor(0.0073) 19862 Training Loss: tensor(0.0078) 19863 Training Loss: tensor(0.0072) 19864 Training Loss: tensor(0.0045) 19865 Training Loss: tensor(0.0031) 19866 Training Loss: tensor(0.0046) 19867 Training Loss: tensor(0.0066) 19868 Training Loss: tensor(0.0058) 19869 Training Loss: tensor(0.0034) 19870 Training Loss: tensor(0.0033) 19871 Training Loss: tensor(0.0031) 19872 Training Loss: tensor(0.0027) 19873 Training Loss: tensor(0.0043) 19874 Training Loss: tensor(0.0029) 19875 Training Loss: tensor(0.0021) 19876 Training Loss: tensor(0.0028) 19877 Training Loss: tensor(0.0021) 19878 Training Loss: tensor(0.0028) 19879 Training Loss: tensor(0.0031) 19880 Training Loss: tensor(0.0021) 19881 Training Loss: tensor(0.0024) 19882 Training Loss: tensor(0.0022) 19883 Training Loss: tensor(0.0019) 19884 Training Loss: tensor(0.0020) 19885 Training Loss: tensor(0.0019) 19886 Training Loss: tensor(0.0016) 19887 Training Loss: tensor(0.0017) 19888 Training Loss: tensor(0.0016) 19889 Training Loss: tensor(0.0020) 19890 Training Loss: tensor(0.0018) 19891 Training Loss: tensor(0.0020) 19892 Training Loss: tensor(0.0026) 19893 Training Loss: tensor(0.0020) 19894 Training Loss: tensor(0.0022) 19895 Training Loss: tensor(0.0018) 19896 Training Loss: tensor(0.0023) 19897 Training Loss: tensor(0.0016) 19898 Training Loss: tensor(0.0021) 19899 Training Loss: tensor(0.0025) 19900 Training Loss: tensor(0.0019) 19901 Training Loss: tensor(0.0015) 19902 Training Loss: tensor(0.0022) 19903 Training Loss: tensor(0.0020) 19904 Training Loss: tensor(0.0016) 19905 Training Loss: tensor(0.0026) 19906 Training Loss: tensor(0.0022) 19907 Training Loss: tensor(0.0020) 19908 Training Loss: tensor(0.0019) 19909 Training Loss: tensor(0.0021) 19910 Training Loss: tensor(0.0023) 19911 Training Loss: tensor(0.0018) 19912 Training Loss: tensor(0.0013) 19913 Training Loss: tensor(0.0018) 19914 Training Loss: tensor(0.0016) 19915 Training Loss: tensor(0.0015) 19916 Training Loss: tensor(0.0020) 19917 Training Loss: tensor(0.0017) 19918 Training Loss: tensor(0.0019) 19919 Training Loss: tensor(0.0019) 19920 Training Loss: tensor(0.0021) 19921 Training Loss: tensor(0.0021) 19922 Training Loss: tensor(0.0017) 19923 Training Loss: tensor(0.0022) 19924 Training Loss: tensor(0.0022) 19925 Training Loss: tensor(0.0017) 19926 Training Loss: tensor(0.0021) 19927 Training Loss: tensor(0.0026) 19928 Training Loss: tensor(0.0018) 19929 Training Loss: tensor(0.0020) 19930 Training Loss: tensor(0.0023) 19931 Training Loss: tensor(0.0017) 19932 Training Loss: tensor(0.0027) 19933 Training Loss: tensor(0.0028) 19934 Training Loss: tensor(0.0023) 19935 Training Loss: tensor(0.0030) 19936 Training Loss: tensor(0.0031) 19937 Training Loss: tensor(0.0024) 19938 Training Loss: tensor(0.0032) 19939 Training Loss: tensor(0.0027) 19940 Training Loss: tensor(0.0021) 19941 Training Loss: tensor(0.0031) 19942 Training Loss: tensor(0.0031) 19943 Training Loss: tensor(0.0021) 19944 Training Loss: tensor(0.0027) 19945 Training Loss: tensor(0.0025) 19946 Training Loss: tensor(0.0017) 19947 Training Loss: tensor(0.0024) 19948 Training Loss: tensor(0.0024) 19949 Training Loss: tensor(0.0026) 19950 Training Loss: tensor(0.0025) 19951 Training Loss: tensor(0.0024) 19952 Training Loss: tensor(0.0024) 19953 Training Loss: tensor(0.0022) 19954 Training Loss: tensor(0.0025) 19955 Training Loss: tensor(0.0024) 19956 Training Loss: tensor(0.0021) 19957 Training Loss: tensor(0.0024) 19958 Training Loss: tensor(0.0028) 19959 Training Loss: tensor(0.0027) 19960 Training Loss: tensor(0.0032) 19961 Training Loss: tensor(0.0043) 19962 Training Loss: tensor(0.0062) 19963 Training Loss: tensor(0.0096) 19964 Training Loss: tensor(0.0129) 19965 Training Loss: tensor(0.0177) 19966 Training Loss: tensor(0.0228) 19967 Training Loss: tensor(0.0287) 19968 Training Loss: tensor(0.0345) 19969 Training Loss: tensor(0.0342) 19970 Training Loss: tensor(0.0291) 19971 Training Loss: tensor(0.0174) 19972 Training Loss: tensor(0.0073) 19973 Training Loss: tensor(0.0019) 19974 Training Loss: tensor(0.0058) 19975 Training Loss: tensor(0.0097) 19976 Training Loss: tensor(0.0118) 19977 Training Loss: tensor(0.0095) 19978 Training Loss: tensor(0.0041) 19979 Training Loss: tensor(0.0023) 19980 Training Loss: tensor(0.0049) 19981 Training Loss: tensor(0.0066) 19982 Training Loss: tensor(0.0054) 19983 Training Loss: tensor(0.0033) 19984 Training Loss: tensor(0.0022) 19985 Training Loss: tensor(0.0041) 19986 Training Loss: tensor(0.0047) 19987 Training Loss: tensor(0.0037) 19988 Training Loss: tensor(0.0044) 19989 Training Loss: tensor(0.0026) 19990 Training Loss: tensor(0.0024) 19991 Training Loss: tensor(0.0054) 19992 Training Loss: tensor(0.0038) 19993 Training Loss: tensor(0.0028) 19994 Training Loss: tensor(0.0025) 19995 Training Loss: tensor(0.0023) 19996 Training Loss: tensor(0.0028) 19997 Training Loss: tensor(0.0029) 19998 Training Loss: tensor(0.0026) 19999 Training Loss: tensor(0.0021)
In [45]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.1, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.2]$¶
- training with iteration 20000
In [46]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.2, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.1483) 1 Training Loss: tensor(0.0968) 2 Training Loss: tensor(0.0992) 3 Training Loss: tensor(0.0725) 4 Training Loss: tensor(0.0659) 5 Training Loss: tensor(0.0862) 6 Training Loss: tensor(0.0655) 7 Training Loss: tensor(0.0755) 8 Training Loss: tensor(0.0644) 9 Training Loss: tensor(0.0601) 10 Training Loss: tensor(0.0458) 11 Training Loss: tensor(0.0361) 12 Training Loss: tensor(0.0359) 13 Training Loss: tensor(0.0330) 14 Training Loss: tensor(0.0371) 15 Training Loss: tensor(0.0353) 16 Training Loss: tensor(0.0324) 17 Training Loss: tensor(0.0360) 18 Training Loss: tensor(0.0236) 19 Training Loss: tensor(0.0261) 20 Training Loss: tensor(0.0210) 21 Training Loss: tensor(0.0199) 22 Training Loss: tensor(0.0199) 23 Training Loss: tensor(0.0170) 24 Training Loss: tensor(0.0186) 25 Training Loss: tensor(0.0230) 26 Training Loss: tensor(0.0214) 27 Training Loss: tensor(0.0190) 28 Training Loss: tensor(0.0150) 29 Training Loss: tensor(0.0180) 30 Training Loss: tensor(0.0163) 31 Training Loss: tensor(0.0155) 32 Training Loss: tensor(0.0175) 33 Training Loss: tensor(0.0174) 34 Training Loss: tensor(0.0228) 35 Training Loss: tensor(0.0175) 36 Training Loss: tensor(0.0157) 37 Training Loss: tensor(0.0174) 38 Training Loss: tensor(0.0168) 39 Training Loss: tensor(0.0152) 40 Training Loss: tensor(0.0139) 41 Training Loss: tensor(0.0127) 42 Training Loss: tensor(0.0150) 43 Training Loss: tensor(0.0149) 44 Training Loss: tensor(0.0139) 45 Training Loss: tensor(0.0159) 46 Training Loss: tensor(0.0123) 47 Training Loss: tensor(0.0139) 48 Training Loss: tensor(0.0135) 49 Training Loss: tensor(0.0101) 50 Training Loss: tensor(0.0131) 51 Training Loss: tensor(0.0145) 52 Training Loss: tensor(0.0114) 53 Training Loss: tensor(0.0148) 54 Training Loss: tensor(0.0103) 55 Training Loss: tensor(0.0116) 56 Training Loss: tensor(0.0152) 57 Training Loss: tensor(0.0107) 58 Training Loss: tensor(0.0129) 59 Training Loss: tensor(0.0113) 60 Training Loss: tensor(0.0139) 61 Training Loss: tensor(0.0094) 62 Training Loss: tensor(0.0128) 63 Training Loss: tensor(0.0123) 64 Training Loss: tensor(0.0102) 65 Training Loss: tensor(0.0112) 66 Training Loss: tensor(0.0104) 67 Training Loss: tensor(0.0134) 68 Training Loss: tensor(0.0108) 69 Training Loss: tensor(0.0092) 70 Training Loss: tensor(0.0120) 71 Training Loss: tensor(0.0106) 72 Training Loss: tensor(0.0136) 73 Training Loss: tensor(0.0117) 74 Training Loss: tensor(0.0093) 75 Training Loss: tensor(0.0112) 76 Training Loss: tensor(0.0108) 77 Training Loss: tensor(0.0094) 78 Training Loss: tensor(0.0117) 79 Training Loss: tensor(0.0104) 80 Training Loss: tensor(0.0099) 81 Training Loss: tensor(0.0113) 82 Training Loss: tensor(0.0101) 83 Training Loss: tensor(0.0108) 84 Training Loss: tensor(0.0108) 85 Training Loss: tensor(0.0092) 86 Training Loss: tensor(0.0111) 87 Training Loss: tensor(0.0097) 88 Training Loss: tensor(0.0092) 89 Training Loss: tensor(0.0100) 90 Training Loss: tensor(0.0109) 91 Training Loss: tensor(0.0086) 92 Training Loss: tensor(0.0116) 93 Training Loss: tensor(0.0098) 94 Training Loss: tensor(0.0105) 95 Training Loss: tensor(0.0104) 96 Training Loss: tensor(0.0096) 97 Training Loss: tensor(0.0101) 98 Training Loss: tensor(0.0101) 99 Training Loss: tensor(0.0085) 100 Training Loss: tensor(0.0116) 101 Training Loss: tensor(0.0095) 102 Training Loss: tensor(0.0116) 103 Training Loss: tensor(0.0116) 104 Training Loss: tensor(0.0104) 105 Training Loss: tensor(0.0102) 106 Training Loss: tensor(0.0101) 107 Training Loss: tensor(0.0085) 108 Training Loss: tensor(0.0115) 109 Training Loss: tensor(0.0109) 110 Training Loss: tensor(0.0097) 111 Training Loss: tensor(0.0096) 112 Training Loss: tensor(0.0096) 113 Training Loss: tensor(0.0111) 114 Training Loss: tensor(0.0089) 115 Training Loss: tensor(0.0099) 116 Training Loss: tensor(0.0091) 117 Training Loss: tensor(0.0106) 118 Training Loss: tensor(0.0097) 119 Training Loss: tensor(0.0074) 120 Training Loss: tensor(0.0096) 121 Training Loss: tensor(0.0084) 122 Training Loss: tensor(0.0099) 123 Training Loss: tensor(0.0099) 124 Training Loss: tensor(0.0102) 125 Training Loss: tensor(0.0096) 126 Training Loss: tensor(0.0096) 127 Training Loss: tensor(0.0081) 128 Training Loss: tensor(0.0091) 129 Training Loss: tensor(0.0102) 130 Training Loss: tensor(0.0089) 131 Training Loss: tensor(0.0090) 132 Training Loss: tensor(0.0082) 133 Training Loss: tensor(0.0086) 134 Training Loss: tensor(0.0091) 135 Training Loss: tensor(0.0071) 136 Training Loss: tensor(0.0101) 137 Training Loss: tensor(0.0086) 138 Training Loss: tensor(0.0090) 139 Training Loss: tensor(0.0096) 140 Training Loss: tensor(0.0083) 141 Training Loss: tensor(0.0099) 142 Training Loss: tensor(0.0100) 143 Training Loss: tensor(0.0073) 144 Training Loss: tensor(0.0078) 145 Training Loss: tensor(0.0076) 146 Training Loss: tensor(0.0096) 147 Training Loss: tensor(0.0075) 148 Training Loss: tensor(0.0079) 149 Training Loss: tensor(0.0088) 150 Training Loss: tensor(0.0091) 151 Training Loss: tensor(0.0087) 152 Training Loss: tensor(0.0081) 153 Training Loss: tensor(0.0108) 154 Training Loss: tensor(0.0086) 155 Training Loss: tensor(0.0092) 156 Training Loss: tensor(0.0085) 157 Training Loss: tensor(0.0077) 158 Training Loss: tensor(0.0068) 159 Training Loss: tensor(0.0085) 160 Training Loss: tensor(0.0078) 161 Training Loss: tensor(0.0073) 162 Training Loss: tensor(0.0073) 163 Training Loss: tensor(0.0083) 164 Training Loss: tensor(0.0076) 165 Training Loss: tensor(0.0099) 166 Training Loss: tensor(0.0076) 167 Training Loss: tensor(0.0074) 168 Training Loss: tensor(0.0075) 169 Training Loss: tensor(0.0085) 170 Training Loss: tensor(0.0080) 171 Training Loss: tensor(0.0089) 172 Training Loss: tensor(0.0079) 173 Training Loss: tensor(0.0078) 174 Training Loss: tensor(0.0079) 175 Training Loss: tensor(0.0082) 176 Training Loss: tensor(0.0090) 177 Training Loss: tensor(0.0074) 178 Training Loss: tensor(0.0086) 179 Training Loss: tensor(0.0084) 180 Training Loss: tensor(0.0089) 181 Training Loss: tensor(0.0077) 182 Training Loss: tensor(0.0077) 183 Training Loss: tensor(0.0068) 184 Training Loss: tensor(0.0080) 185 Training Loss: tensor(0.0070) 186 Training Loss: tensor(0.0078) 187 Training Loss: tensor(0.0062) 188 Training Loss: tensor(0.0076) 189 Training Loss: tensor(0.0068) 190 Training Loss: tensor(0.0061) 191 Training Loss: tensor(0.0061) 192 Training Loss: tensor(0.0068) 193 Training Loss: tensor(0.0063) 194 Training Loss: tensor(0.0078) 195 Training Loss: tensor(0.0084) 196 Training Loss: tensor(0.0064) 197 Training Loss: tensor(0.0079) 198 Training Loss: tensor(0.0070) 199 Training Loss: tensor(0.0087) 200 Training Loss: tensor(0.0067) 201 Training Loss: tensor(0.0073) 202 Training Loss: tensor(0.0065) 203 Training Loss: tensor(0.0079) 204 Training Loss: tensor(0.0066) 205 Training Loss: tensor(0.0073) 206 Training Loss: tensor(0.0091) 207 Training Loss: tensor(0.0069) 208 Training Loss: tensor(0.0090) 209 Training Loss: tensor(0.0075) 210 Training Loss: tensor(0.0106) 211 Training Loss: tensor(0.0071) 212 Training Loss: tensor(0.0093) 213 Training Loss: tensor(0.0078) 214 Training Loss: tensor(0.0088) 215 Training Loss: tensor(0.0080) 216 Training Loss: tensor(0.0073) 217 Training Loss: tensor(0.0063) 218 Training Loss: tensor(0.0070) 219 Training Loss: tensor(0.0081) 220 Training Loss: tensor(0.0076) 221 Training Loss: tensor(0.0069) 222 Training Loss: tensor(0.0068) 223 Training Loss: tensor(0.0082) 224 Training Loss: tensor(0.0074) 225 Training Loss: tensor(0.0075) 226 Training Loss: tensor(0.0074) 227 Training Loss: tensor(0.0086) 228 Training Loss: tensor(0.0075) 229 Training Loss: tensor(0.0078) 230 Training Loss: tensor(0.0070) 231 Training Loss: tensor(0.0074) 232 Training Loss: tensor(0.0073) 233 Training Loss: tensor(0.0059) 234 Training Loss: tensor(0.0082) 235 Training Loss: tensor(0.0070) 236 Training Loss: tensor(0.0071) 237 Training Loss: tensor(0.0065) 238 Training Loss: tensor(0.0067) 239 Training Loss: tensor(0.0071) 240 Training Loss: tensor(0.0074) 241 Training Loss: tensor(0.0069) 242 Training Loss: tensor(0.0078) 243 Training Loss: tensor(0.0065) 244 Training Loss: tensor(0.0072) 245 Training Loss: tensor(0.0091) 246 Training Loss: tensor(0.0071) 247 Training Loss: tensor(0.0079) 248 Training Loss: tensor(0.0056) 249 Training Loss: tensor(0.0076) 250 Training Loss: tensor(0.0069) 251 Training Loss: tensor(0.0069) 252 Training Loss: tensor(0.0068) 253 Training Loss: tensor(0.0076) 254 Training Loss: tensor(0.0066) 255 Training Loss: tensor(0.0063) 256 Training Loss: tensor(0.0061) 257 Training Loss: tensor(0.0065) 258 Training Loss: tensor(0.0084) 259 Training Loss: tensor(0.0060) 260 Training Loss: tensor(0.0065) 261 Training Loss: tensor(0.0067) 262 Training Loss: tensor(0.0054) 263 Training Loss: tensor(0.0063) 264 Training Loss: tensor(0.0065) 265 Training Loss: tensor(0.0062) 266 Training Loss: tensor(0.0059) 267 Training Loss: tensor(0.0061) 268 Training Loss: tensor(0.0054) 269 Training Loss: tensor(0.0065) 270 Training Loss: tensor(0.0060) 271 Training Loss: tensor(0.0057) 272 Training Loss: tensor(0.0069) 273 Training Loss: tensor(0.0069) 274 Training Loss: tensor(0.0066) 275 Training Loss: tensor(0.0062) 276 Training Loss: tensor(0.0061) 277 Training Loss: tensor(0.0066) 278 Training Loss: tensor(0.0083) 279 Training Loss: tensor(0.0062) 280 Training Loss: tensor(0.0062) 281 Training Loss: tensor(0.0071) 282 Training Loss: tensor(0.0069) 283 Training Loss: tensor(0.0070) 284 Training Loss: tensor(0.0071) 285 Training Loss: tensor(0.0064) 286 Training Loss: tensor(0.0074) 287 Training Loss: tensor(0.0086) 288 Training Loss: tensor(0.0071) 289 Training Loss: tensor(0.0068) 290 Training Loss: tensor(0.0069) 291 Training Loss: tensor(0.0065) 292 Training Loss: tensor(0.0066) 293 Training Loss: tensor(0.0076) 294 Training Loss: tensor(0.0069) 295 Training Loss: tensor(0.0066) 296 Training 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In [47]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.2, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.3]$¶
- training with iteration 20000
In [48]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.3, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0027) 1 Training Loss: tensor(0.0019) 2 Training Loss: tensor(0.0021) 3 Training Loss: tensor(0.0017) 4 Training Loss: tensor(0.0018) 5 Training Loss: tensor(0.0016) 6 Training Loss: tensor(0.0019) 7 Training Loss: tensor(0.0019) 8 Training Loss: tensor(0.0021) 9 Training Loss: tensor(0.0016) 10 Training Loss: tensor(0.0017) 11 Training Loss: tensor(0.0014) 12 Training Loss: tensor(0.0020) 13 Training Loss: tensor(0.0014) 14 Training Loss: tensor(0.0016) 15 Training Loss: tensor(0.0016) 16 Training Loss: tensor(0.0015) 17 Training Loss: tensor(0.0013) 18 Training Loss: tensor(0.0015) 19 Training Loss: tensor(0.0013) 20 Training Loss: tensor(0.0014) 21 Training Loss: tensor(0.0011) 22 Training Loss: tensor(0.0014) 23 Training Loss: tensor(0.0010) 24 Training Loss: tensor(0.0013) 25 Training Loss: tensor(0.0010) 26 Training Loss: tensor(0.0009) 27 Training Loss: tensor(0.0008) 28 Training Loss: tensor(0.0009) 29 Training Loss: tensor(0.0008) 30 Training Loss: tensor(0.0007) 31 Training Loss: tensor(0.0007) 32 Training Loss: tensor(0.0007) 33 Training Loss: tensor(0.0007) 34 Training Loss: tensor(0.0007) 35 Training Loss: tensor(0.0008) 36 Training Loss: tensor(0.0007) 37 Training Loss: tensor(0.0008) 38 Training Loss: tensor(0.0009) 39 Training Loss: tensor(0.0009) 40 Training Loss: tensor(0.0008) 41 Training Loss: tensor(0.0008) 42 Training Loss: tensor(0.0008) 43 Training Loss: tensor(0.0009) 44 Training Loss: tensor(0.0007) 45 Training Loss: tensor(0.0009) 46 Training Loss: tensor(0.0008) 47 Training Loss: tensor(0.0007) 48 Training Loss: tensor(0.0007) 49 Training Loss: tensor(0.0005) 50 Training Loss: tensor(0.0008) 51 Training Loss: tensor(0.0006) 52 Training Loss: tensor(0.0007) 53 Training Loss: tensor(0.0008) 54 Training Loss: tensor(0.0007) 55 Training Loss: tensor(0.0007) 56 Training Loss: tensor(0.0005) 57 Training Loss: tensor(0.0007) 58 Training Loss: tensor(0.0006) 59 Training Loss: tensor(0.0006) 60 Training Loss: tensor(0.0006) 61 Training Loss: tensor(0.0005) 62 Training Loss: tensor(0.0008) 63 Training Loss: tensor(0.0005) 64 Training Loss: tensor(0.0007) 65 Training Loss: tensor(0.0005) 66 Training Loss: tensor(0.0006) 67 Training Loss: tensor(0.0006) 68 Training Loss: tensor(0.0005) 69 Training Loss: tensor(0.0006) 70 Training Loss: tensor(0.0006) 71 Training Loss: tensor(0.0006) 72 Training Loss: tensor(0.0007) 73 Training Loss: tensor(0.0007) 74 Training Loss: tensor(0.0005) 75 Training Loss: tensor(0.0006) 76 Training Loss: tensor(0.0005) 77 Training Loss: tensor(0.0006) 78 Training Loss: tensor(0.0006) 79 Training Loss: tensor(0.0009) 80 Training Loss: tensor(0.0004) 81 Training Loss: tensor(0.0007) 82 Training Loss: tensor(0.0005) 83 Training Loss: tensor(0.0007) 84 Training Loss: tensor(0.0005) 85 Training Loss: tensor(0.0006) 86 Training Loss: tensor(0.0007) 87 Training Loss: tensor(0.0006) 88 Training Loss: tensor(0.0006) 89 Training Loss: tensor(0.0007) 90 Training Loss: tensor(0.0005) 91 Training Loss: tensor(0.0006) 92 Training Loss: tensor(0.0006) 93 Training Loss: tensor(0.0005) 94 Training Loss: tensor(0.0005) 95 Training Loss: tensor(0.0005) 96 Training Loss: tensor(0.0006) 97 Training Loss: tensor(0.0005) 98 Training Loss: tensor(0.0006) 99 Training Loss: tensor(0.0005) 100 Training Loss: tensor(0.0005) 101 Training Loss: tensor(0.0004) 102 Training Loss: tensor(0.0004) 103 Training Loss: tensor(0.0005) 104 Training Loss: tensor(0.0006) 105 Training Loss: tensor(0.0006) 106 Training Loss: tensor(0.0005) 107 Training Loss: tensor(0.0005) 108 Training Loss: tensor(0.0005) 109 Training Loss: tensor(0.0004) 110 Training Loss: tensor(0.0005) 111 Training Loss: tensor(0.0006) 112 Training Loss: tensor(0.0007) 113 Training Loss: tensor(0.0006) 114 Training Loss: tensor(0.0007) 115 Training Loss: tensor(0.0007) 116 Training Loss: tensor(0.0007) 117 Training Loss: tensor(0.0006) 118 Training Loss: tensor(0.0006) 119 Training Loss: tensor(0.0005) 120 Training Loss: tensor(0.0005) 121 Training Loss: tensor(0.0005) 122 Training Loss: tensor(0.0005) 123 Training Loss: tensor(0.0006) 124 Training Loss: tensor(0.0005) 125 Training Loss: tensor(0.0007) 126 Training Loss: tensor(0.0006) 127 Training Loss: tensor(0.0005) 128 Training Loss: tensor(0.0006) 129 Training Loss: tensor(0.0005) 130 Training Loss: tensor(0.0005) 131 Training Loss: tensor(0.0005) 132 Training Loss: tensor(0.0005) 133 Training Loss: tensor(0.0005) 134 Training Loss: tensor(0.0004) 135 Training Loss: tensor(0.0005) 136 Training Loss: tensor(0.0006) 137 Training Loss: tensor(0.0008) 138 Training Loss: tensor(0.0009) 139 Training Loss: tensor(0.0010) 140 Training Loss: tensor(0.0013) 141 Training Loss: tensor(0.0019) 142 Training Loss: tensor(0.0025) 143 Training Loss: tensor(0.0031) 144 Training Loss: tensor(0.0033) 145 Training Loss: tensor(0.0028) 146 Training Loss: tensor(0.0024) 147 Training Loss: tensor(0.0015) 148 Training Loss: tensor(0.0010) 149 Training Loss: tensor(0.0005) 150 Training Loss: tensor(0.0005) 151 Training Loss: tensor(0.0007) 152 Training Loss: tensor(0.0010) 153 Training Loss: tensor(0.0014) 154 Training Loss: tensor(0.0014) 155 Training Loss: tensor(0.0014) 156 Training Loss: tensor(0.0011) 157 Training Loss: tensor(0.0008) 158 Training Loss: tensor(0.0005) 159 Training Loss: tensor(0.0004) 160 Training Loss: tensor(0.0007) 161 Training Loss: tensor(0.0009) 162 Training Loss: tensor(0.0011) 163 Training Loss: tensor(0.0012) 164 Training Loss: tensor(0.0010) 165 Training Loss: tensor(0.0008) 166 Training Loss: tensor(0.0006) 167 Training Loss: tensor(0.0004) 168 Training Loss: tensor(0.0004) 169 Training Loss: tensor(0.0005) 170 Training Loss: tensor(0.0006) 171 Training Loss: tensor(0.0007) 172 Training Loss: tensor(0.0007) 173 Training Loss: tensor(0.0007) 174 Training Loss: tensor(0.0007) 175 Training Loss: tensor(0.0007) 176 Training Loss: tensor(0.0007) 177 Training Loss: tensor(0.0006) 178 Training Loss: tensor(0.0007) 179 Training Loss: tensor(0.0005) 180 Training Loss: tensor(0.0005) 181 Training Loss: tensor(0.0005) 182 Training Loss: tensor(0.0004) 183 Training Loss: tensor(0.0005) 184 Training Loss: tensor(0.0005) 185 Training Loss: tensor(0.0005) 186 Training Loss: tensor(0.0005) 187 Training Loss: tensor(0.0005) 188 Training Loss: tensor(0.0005) 189 Training Loss: tensor(0.0005) 190 Training Loss: tensor(0.0005) 191 Training Loss: tensor(0.0005) 192 Training Loss: tensor(0.0004) 193 Training Loss: tensor(0.0005) 194 Training Loss: tensor(0.0006) 195 Training Loss: tensor(0.0006) 196 Training Loss: tensor(0.0007) 197 Training Loss: tensor(0.0009) 198 Training Loss: tensor(0.0012) 199 Training Loss: tensor(0.0013) 200 Training Loss: tensor(0.0014) 201 Training Loss: tensor(0.0014) 202 Training Loss: tensor(0.0012) 203 Training Loss: tensor(0.0010) 204 Training Loss: tensor(0.0008) 205 Training Loss: tensor(0.0007) 206 Training Loss: tensor(0.0008) 207 Training Loss: tensor(0.0008) 208 Training Loss: tensor(0.0004) 209 Training Loss: tensor(0.0008) 210 Training Loss: tensor(0.0010) 211 Training Loss: tensor(0.0016) 212 Training Loss: tensor(0.0020) 213 Training Loss: tensor(0.0024) 214 Training Loss: tensor(0.0025) 215 Training Loss: tensor(0.0025) 216 Training Loss: tensor(0.0022) 217 Training Loss: tensor(0.0017) 218 Training Loss: tensor(0.0011) 219 Training Loss: tensor(0.0007) 220 Training Loss: tensor(0.0004) 221 Training Loss: tensor(0.0007) 222 Training Loss: tensor(0.0010) 223 Training Loss: tensor(0.0017) 224 Training Loss: tensor(0.0023) 225 Training Loss: tensor(0.0029) 226 Training Loss: tensor(0.0029) 227 Training Loss: tensor(0.0028) 228 Training Loss: tensor(0.0022) 229 Training Loss: tensor(0.0017) 230 Training Loss: tensor(0.0009) 231 Training Loss: tensor(0.0005) 232 Training Loss: tensor(0.0006) 233 Training Loss: tensor(0.0010) 234 Training Loss: tensor(0.0014) 235 Training Loss: tensor(0.0018) 236 Training Loss: tensor(0.0017) 237 Training Loss: tensor(0.0016) 238 Training Loss: tensor(0.0011) 239 Training Loss: tensor(0.0007) 240 Training Loss: tensor(0.0005) 241 Training Loss: tensor(0.0004) 242 Training Loss: tensor(0.0008) 243 Training Loss: tensor(0.0011) 244 Training Loss: tensor(0.0014) 245 Training Loss: tensor(0.0016) 246 Training Loss: tensor(0.0014) 247 Training Loss: tensor(0.0013) 248 Training Loss: tensor(0.0009) 249 Training Loss: tensor(0.0006) 250 Training Loss: tensor(0.0004) 251 Training Loss: tensor(0.0005) 252 Training Loss: tensor(0.0005) 253 Training Loss: tensor(0.0004) 254 Training Loss: tensor(0.0006) 255 Training Loss: tensor(0.0005) 256 Training Loss: tensor(0.0006) 257 Training Loss: tensor(0.0005) 258 Training Loss: tensor(0.0003) 259 Training Loss: tensor(0.0005) 260 Training Loss: tensor(0.0004) 261 Training Loss: tensor(0.0003) 262 Training Loss: tensor(0.0004) 263 Training Loss: tensor(0.0004) 264 Training Loss: tensor(0.0004) 265 Training Loss: tensor(0.0005) 266 Training Loss: tensor(0.0004) 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355 Training Loss: tensor(0.0029) 356 Training Loss: tensor(0.0027) 357 Training Loss: tensor(0.0014) 358 Training Loss: tensor(0.0006) 359 Training Loss: tensor(0.0004) 360 Training Loss: tensor(0.0008) 361 Training Loss: tensor(0.0009) 362 Training Loss: tensor(0.0008) 363 Training Loss: tensor(0.0006) 364 Training Loss: tensor(0.0004) 365 Training Loss: tensor(0.0004) 366 Training Loss: tensor(0.0005) 367 Training Loss: tensor(0.0006) 368 Training Loss: tensor(0.0007) 369 Training Loss: tensor(0.0004) 370 Training Loss: tensor(0.0005) 371 Training Loss: tensor(0.0004) 372 Training Loss: tensor(0.0005) 373 Training Loss: tensor(0.0004) 374 Training Loss: tensor(0.0005) 375 Training Loss: tensor(0.0004) 376 Training Loss: tensor(0.0003) 377 Training Loss: tensor(0.0004) 378 Training Loss: tensor(0.0004) 379 Training Loss: tensor(0.0004) 380 Training Loss: tensor(0.0004) 381 Training Loss: tensor(0.0004) 382 Training Loss: tensor(0.0003) 383 Training Loss: tensor(0.0004) 384 Training 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531 Training Loss: tensor(0.0014) 532 Training Loss: tensor(0.0015) 533 Training Loss: tensor(0.0012) 534 Training Loss: tensor(0.0011) 535 Training Loss: tensor(0.0008) 536 Training Loss: tensor(0.0005) 537 Training Loss: tensor(0.0005) 538 Training Loss: tensor(0.0005) 539 Training Loss: tensor(0.0007) 540 Training Loss: tensor(0.0009) 541 Training Loss: tensor(0.0009) 542 Training Loss: tensor(0.0009) 543 Training Loss: tensor(0.0009) 544 Training Loss: tensor(0.0008) 545 Training Loss: tensor(0.0007) 546 Training Loss: tensor(0.0005) 547 Training Loss: tensor(0.0005) 548 Training Loss: tensor(0.0005) 549 Training Loss: tensor(0.0003) 550 Training Loss: tensor(0.0004) 551 Training Loss: tensor(0.0003) 552 Training Loss: tensor(0.0004) 553 Training Loss: tensor(0.0004) 554 Training Loss: tensor(0.0003) 555 Training Loss: tensor(0.0004) 556 Training Loss: tensor(0.0004) 557 Training Loss: tensor(0.0005) 558 Training Loss: tensor(0.0005) 559 Training Loss: tensor(0.0005) 560 Training 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619 Training Loss: tensor(0.0007) 620 Training Loss: tensor(0.0006) 621 Training Loss: tensor(0.0013) 622 Training Loss: tensor(0.0021) 623 Training Loss: tensor(0.0029) 624 Training Loss: tensor(0.0028) 625 Training Loss: tensor(0.0022) 626 Training Loss: tensor(0.0012) 627 Training Loss: tensor(0.0008) 628 Training Loss: tensor(0.0006) 629 Training Loss: tensor(0.0007) 630 Training Loss: tensor(0.0012) 631 Training Loss: tensor(0.0015) 632 Training Loss: tensor(0.0014) 633 Training Loss: tensor(0.0012) 634 Training Loss: tensor(0.0007) 635 Training Loss: tensor(0.0007) 636 Training Loss: tensor(0.0013) 637 Training Loss: tensor(0.0015) 638 Training Loss: tensor(0.0016) 639 Training Loss: tensor(0.0015) 640 Training Loss: tensor(0.0008) 641 Training Loss: tensor(0.0006) 642 Training Loss: tensor(0.0007) 643 Training Loss: tensor(0.0010) 644 Training Loss: tensor(0.0014) 645 Training Loss: tensor(0.0013) 646 Training Loss: tensor(0.0011) 647 Training Loss: tensor(0.0009) 648 Training 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Training Loss: tensor(0.0010) 19877 Training Loss: tensor(0.0006) 19878 Training Loss: tensor(0.0008) 19879 Training Loss: tensor(0.0006) 19880 Training Loss: tensor(0.0008) 19881 Training Loss: tensor(0.0006) 19882 Training Loss: tensor(0.0005) 19883 Training Loss: tensor(0.0008) 19884 Training Loss: tensor(0.0003) 19885 Training Loss: tensor(0.0010) 19886 Training Loss: tensor(0.0007) 19887 Training Loss: tensor(0.0004) 19888 Training Loss: tensor(0.0005) 19889 Training Loss: tensor(0.0005) 19890 Training Loss: tensor(0.0012) 19891 Training Loss: tensor(0.0006) 19892 Training Loss: tensor(0.0013) 19893 Training Loss: tensor(0.0008) 19894 Training Loss: tensor(0.0012) 19895 Training Loss: tensor(0.0009) 19896 Training Loss: tensor(0.0010) 19897 Training Loss: tensor(0.0013) 19898 Training Loss: tensor(0.0007) 19899 Training Loss: tensor(0.0011) 19900 Training Loss: tensor(0.0014) 19901 Training Loss: tensor(0.0013) 19902 Training Loss: tensor(0.0013) 19903 Training Loss: tensor(0.0013) 19904 Training Loss: tensor(0.0009) 19905 Training Loss: tensor(0.0013) 19906 Training Loss: tensor(0.0015) 19907 Training Loss: tensor(0.0006) 19908 Training Loss: tensor(0.0011) 19909 Training Loss: tensor(0.0007) 19910 Training Loss: tensor(0.0011) 19911 Training Loss: tensor(0.0006) 19912 Training Loss: tensor(0.0007) 19913 Training Loss: tensor(0.0009) 19914 Training Loss: tensor(0.0008) 19915 Training Loss: tensor(0.0006) 19916 Training Loss: tensor(0.0008) 19917 Training Loss: tensor(0.0003) 19918 Training Loss: tensor(0.0009) 19919 Training Loss: tensor(0.0006) 19920 Training Loss: tensor(0.0010) 19921 Training Loss: tensor(0.0014) 19922 Training Loss: tensor(0.0011) 19923 Training Loss: tensor(0.0011) 19924 Training Loss: tensor(0.0004) 19925 Training Loss: tensor(0.0005) 19926 Training Loss: tensor(0.0007) 19927 Training Loss: tensor(0.0011) 19928 Training Loss: tensor(0.0011) 19929 Training Loss: tensor(0.0011) 19930 Training Loss: tensor(0.0009) 19931 Training Loss: tensor(0.0007) 19932 Training Loss: tensor(0.0003) 19933 Training Loss: tensor(0.0005) 19934 Training Loss: tensor(0.0003) 19935 Training Loss: tensor(0.0004) 19936 Training Loss: tensor(0.0005) 19937 Training Loss: tensor(0.0003) 19938 Training Loss: tensor(0.0003) 19939 Training Loss: tensor(0.0001) 19940 Training Loss: tensor(0.0003) 19941 Training Loss: tensor(0.0003) 19942 Training Loss: tensor(0.0003) 19943 Training Loss: tensor(0.0002) 19944 Training Loss: tensor(0.0003) 19945 Training Loss: tensor(0.0002) 19946 Training Loss: tensor(0.0003) 19947 Training Loss: tensor(0.0002) 19948 Training Loss: tensor(0.0003) 19949 Training Loss: tensor(0.0002) 19950 Training Loss: tensor(0.0002) 19951 Training Loss: tensor(0.0002) 19952 Training Loss: tensor(0.0002) 19953 Training Loss: tensor(0.0002) 19954 Training Loss: tensor(0.0002) 19955 Training Loss: tensor(0.0002) 19956 Training Loss: tensor(0.0002) 19957 Training Loss: tensor(0.0002) 19958 Training Loss: tensor(0.0002) 19959 Training Loss: tensor(0.0002) 19960 Training Loss: tensor(0.0002) 19961 Training Loss: tensor(0.0004) 19962 Training Loss: tensor(0.0005) 19963 Training Loss: tensor(0.0006) 19964 Training Loss: tensor(0.0008) 19965 Training Loss: tensor(0.0006) 19966 Training Loss: tensor(0.0004) 19967 Training Loss: tensor(0.0002) 19968 Training Loss: tensor(0.0002) 19969 Training Loss: tensor(0.0003) 19970 Training Loss: tensor(0.0004) 19971 Training Loss: tensor(0.0004) 19972 Training Loss: tensor(0.0002) 19973 Training Loss: tensor(0.0002) 19974 Training Loss: tensor(0.0003) 19975 Training Loss: tensor(0.0004) 19976 Training Loss: tensor(0.0007) 19977 Training Loss: tensor(0.0006) 19978 Training Loss: tensor(0.0006) 19979 Training Loss: tensor(0.0003) 19980 Training Loss: tensor(0.0002) 19981 Training Loss: tensor(0.0005) 19982 Training Loss: tensor(0.0005) 19983 Training Loss: tensor(0.0007) 19984 Training Loss: tensor(0.0011) 19985 Training Loss: tensor(0.0008) 19986 Training Loss: tensor(0.0009) 19987 Training Loss: tensor(0.0013) 19988 Training Loss: tensor(0.0008) 19989 Training Loss: tensor(0.0008) 19990 Training Loss: tensor(0.0007) 19991 Training Loss: tensor(0.0003) 19992 Training Loss: tensor(0.0004) 19993 Training Loss: tensor(0.0002) 19994 Training Loss: tensor(0.0005) 19995 Training Loss: tensor(0.0007) 19996 Training Loss: tensor(0.0006) 19997 Training Loss: tensor(0.0007) 19998 Training Loss: tensor(0.0004) 19999 Training Loss: tensor(0.0004)
In [49]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.3, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.4]$¶
- training with iteration 20000
In [50]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.4, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0005) 1 Training Loss: tensor(0.0004) 2 Training Loss: tensor(0.0006) 3 Training Loss: tensor(0.0003) 4 Training Loss: tensor(0.0005) 5 Training Loss: tensor(0.0004) 6 Training Loss: tensor(0.0004) 7 Training Loss: tensor(0.0004) 8 Training Loss: tensor(0.0004) 9 Training Loss: tensor(0.0006) 10 Training Loss: tensor(0.0006) 11 Training Loss: tensor(0.0004) 12 Training Loss: tensor(0.0005) 13 Training Loss: tensor(0.0004) 14 Training Loss: tensor(0.0005) 15 Training Loss: tensor(0.0005) 16 Training Loss: tensor(0.0004) 17 Training Loss: tensor(0.0003) 18 Training Loss: tensor(0.0003) 19 Training Loss: tensor(0.0004) 20 Training Loss: tensor(0.0005) 21 Training Loss: tensor(0.0006) 22 Training Loss: tensor(0.0004) 23 Training Loss: tensor(0.0003) 24 Training Loss: tensor(0.0004) 25 Training Loss: tensor(0.0003) 26 Training Loss: tensor(0.0005) 27 Training Loss: tensor(0.0004) 28 Training Loss: tensor(0.0006) 29 Training Loss: tensor(0.0002) 30 Training Loss: tensor(0.0005) 31 Training Loss: tensor(0.0002) 32 Training Loss: tensor(0.0004) 33 Training Loss: tensor(0.0003) 34 Training Loss: tensor(0.0004) 35 Training Loss: tensor(0.0003) 36 Training Loss: tensor(0.0003) 37 Training Loss: tensor(0.0004) 38 Training Loss: tensor(0.0003) 39 Training Loss: tensor(0.0004) 40 Training Loss: tensor(0.0002) 41 Training Loss: tensor(0.0004) 42 Training Loss: tensor(0.0003) 43 Training Loss: tensor(0.0004) 44 Training Loss: tensor(0.0003) 45 Training Loss: tensor(0.0003) 46 Training Loss: tensor(0.0004) 47 Training Loss: tensor(0.0003) 48 Training Loss: tensor(0.0003) 49 Training Loss: tensor(0.0004) 50 Training Loss: tensor(0.0003) 51 Training Loss: tensor(0.0003) 52 Training Loss: tensor(0.0003) 53 Training Loss: tensor(0.0002) 54 Training Loss: tensor(0.0002) 55 Training Loss: tensor(0.0002) 56 Training Loss: tensor(0.0002) 57 Training Loss: tensor(0.0002) 58 Training Loss: tensor(0.0002) 59 Training Loss: tensor(0.0001) 60 Training Loss: tensor(0.0002) 61 Training Loss: tensor(0.0002) 62 Training Loss: tensor(0.0002) 63 Training Loss: tensor(0.0002) 64 Training Loss: tensor(0.0002) 65 Training Loss: tensor(0.0002) 66 Training Loss: tensor(0.0002) 67 Training Loss: tensor(0.0002) 68 Training Loss: tensor(0.0002) 69 Training Loss: tensor(0.0002) 70 Training Loss: tensor(0.0002) 71 Training Loss: tensor(0.0003) 72 Training Loss: tensor(0.0002) 73 Training Loss: tensor(0.0003) 74 Training Loss: tensor(0.0002) 75 Training Loss: tensor(0.0002) 76 Training Loss: tensor(0.0002) 77 Training Loss: tensor(0.0002) 78 Training Loss: tensor(0.0003) 79 Training Loss: tensor(0.0001) 80 Training Loss: tensor(0.0002) 81 Training Loss: tensor(0.0002) 82 Training Loss: tensor(0.0002) 83 Training Loss: tensor(0.0002) 84 Training Loss: tensor(0.0002) 85 Training Loss: tensor(0.0002) 86 Training Loss: tensor(0.0002) 87 Training Loss: tensor(0.0001) 88 Training Loss: tensor(0.0002) 89 Training Loss: tensor(0.0001) 90 Training Loss: tensor(0.0002) 91 Training Loss: tensor(0.0002) 92 Training Loss: tensor(0.0002) 93 Training Loss: tensor(0.0002) 94 Training Loss: tensor(0.0001) 95 Training Loss: tensor(0.0002) 96 Training Loss: tensor(0.0002) 97 Training Loss: tensor(0.0003) 98 Training Loss: tensor(0.0003) 99 Training Loss: tensor(0.0005) 100 Training Loss: tensor(0.0004) 101 Training Loss: tensor(0.0004) 102 Training Loss: tensor(0.0004) 103 Training Loss: tensor(0.0002) 104 Training Loss: tensor(0.0002) 105 Training Loss: tensor(0.0002) 106 Training Loss: tensor(0.0002) 107 Training Loss: tensor(0.0002) 108 Training Loss: tensor(0.0003) 109 Training Loss: tensor(0.0002) 110 Training Loss: tensor(0.0002) 111 Training Loss: tensor(0.0002) 112 Training Loss: tensor(0.0002) 113 Training Loss: tensor(0.0002) 114 Training Loss: tensor(0.0002) 115 Training Loss: tensor(0.0002) 116 Training Loss: tensor(0.0002) 117 Training Loss: tensor(0.0002) 118 Training Loss: tensor(0.0001) 119 Training Loss: tensor(0.0003) 120 Training Loss: tensor(0.0002) 121 Training Loss: tensor(0.0004) 122 Training Loss: tensor(0.0003) 123 Training Loss: tensor(0.0002) 124 Training Loss: tensor(0.0002) 125 Training Loss: tensor(0.0002) 126 Training Loss: tensor(0.0002) 127 Training Loss: tensor(0.0002) 128 Training Loss: tensor(0.0003) 129 Training Loss: tensor(0.0003) 130 Training Loss: tensor(0.0003) 131 Training Loss: tensor(0.0002) 132 Training Loss: tensor(0.0002) 133 Training Loss: tensor(0.0001) 134 Training Loss: tensor(0.0002) 135 Training Loss: tensor(0.0002) 136 Training Loss: tensor(0.0002) 137 Training Loss: tensor(0.0001) 138 Training Loss: tensor(0.0002) 139 Training Loss: tensor(0.0001) 140 Training Loss: tensor(0.0002) 141 Training Loss: tensor(0.0001) 142 Training Loss: tensor(0.0002) 143 Training Loss: tensor(0.0001) 144 Training Loss: tensor(0.0001) 145 Training Loss: tensor(0.0001) 146 Training Loss: tensor(0.0001) 147 Training Loss: tensor(0.0002) 148 Training Loss: tensor(0.0002) 149 Training Loss: tensor(0.0002) 150 Training Loss: tensor(0.0002) 151 Training Loss: tensor(0.0002) 152 Training Loss: tensor(0.0002) 153 Training Loss: tensor(0.0002) 154 Training Loss: tensor(0.0001) 155 Training Loss: tensor(0.0002) 156 Training Loss: tensor(0.0001) 157 Training Loss: tensor(0.0002) 158 Training Loss: tensor(0.0002) 159 Training Loss: tensor(0.0002) 160 Training Loss: tensor(0.0002) 161 Training Loss: tensor(0.0002) 162 Training Loss: tensor(0.0002) 163 Training Loss: tensor(0.0002) 164 Training Loss: tensor(0.0002) 165 Training Loss: tensor(0.0002) 166 Training Loss: tensor(0.0003) 167 Training Loss: tensor(0.0002) 168 Training Loss: tensor(0.0002) 169 Training Loss: tensor(0.0003) 170 Training Loss: tensor(0.0002) 171 Training Loss: tensor(0.0002) 172 Training Loss: tensor(0.0002) 173 Training Loss: tensor(0.0001) 174 Training Loss: tensor(0.0003) 175 Training Loss: tensor(0.0002) 176 Training Loss: tensor(0.0003) 177 Training Loss: tensor(0.0003) 178 Training Loss: tensor(0.0003) 179 Training Loss: tensor(0.0002) 180 Training Loss: tensor(0.0003) 181 Training Loss: tensor(0.0002) 182 Training Loss: tensor(0.0002) 183 Training Loss: tensor(0.0002) 184 Training Loss: tensor(0.0002) 185 Training Loss: tensor(0.0001) 186 Training Loss: tensor(0.0002) 187 Training Loss: tensor(0.0001) 188 Training Loss: tensor(0.0002) 189 Training Loss: tensor(0.0002) 190 Training Loss: tensor(0.0002) 191 Training Loss: tensor(0.0001) 192 Training Loss: tensor(0.0002) 193 Training Loss: tensor(0.0001) 194 Training Loss: tensor(0.0002) 195 Training Loss: tensor(0.0001) 196 Training Loss: tensor(0.0002) 197 Training Loss: tensor(0.0002) 198 Training Loss: tensor(0.0002) 199 Training Loss: tensor(0.0002) 200 Training Loss: tensor(0.0002) 201 Training Loss: tensor(0.0002) 202 Training Loss: tensor(0.0002) 203 Training Loss: tensor(0.0002) 204 Training Loss: tensor(0.0002) 205 Training Loss: tensor(0.0002) 206 Training Loss: tensor(0.0001) 207 Training Loss: tensor(0.0002) 208 Training Loss: tensor(0.0002) 209 Training Loss: tensor(0.0001) 210 Training Loss: tensor(0.0001) 211 Training Loss: tensor(0.0002) 212 Training Loss: tensor(0.0001) 213 Training Loss: tensor(0.0002) 214 Training Loss: tensor(0.0001) 215 Training Loss: tensor(0.0002) 216 Training Loss: tensor(0.0001) 217 Training Loss: tensor(0.0002) 218 Training Loss: tensor(0.0001) 219 Training Loss: tensor(0.0002) 220 Training Loss: tensor(0.0002) 221 Training Loss: tensor(0.0002) 222 Training Loss: tensor(0.0002) 223 Training Loss: tensor(0.0003) 224 Training Loss: tensor(0.0002) 225 Training Loss: tensor(0.0002) 226 Training Loss: tensor(0.0002) 227 Training Loss: tensor(0.0002) 228 Training Loss: tensor(0.0002) 229 Training Loss: tensor(0.0001) 230 Training Loss: tensor(0.0001) 231 Training Loss: tensor(0.0001) 232 Training Loss: tensor(0.0001) 233 Training Loss: tensor(0.0001) 234 Training Loss: tensor(0.0001) 235 Training Loss: tensor(0.0001) 236 Training Loss: tensor(0.0002) 237 Training Loss: tensor(0.0002) 238 Training Loss: tensor(0.0004) 239 Training Loss: tensor(0.0005) 240 Training Loss: tensor(0.0009) 241 Training Loss: tensor(0.0011) 242 Training Loss: tensor(0.0015) 243 Training Loss: tensor(0.0019) 244 Training Loss: tensor(0.0021) 245 Training Loss: tensor(0.0024) 246 Training Loss: tensor(0.0024) 247 Training Loss: tensor(0.0022) 248 Training Loss: tensor(0.0019) 249 Training Loss: tensor(0.0012) 250 Training Loss: tensor(0.0005) 251 Training Loss: tensor(0.0002) 252 Training Loss: tensor(0.0003) 253 Training Loss: tensor(0.0005) 254 Training Loss: tensor(0.0005) 255 Training Loss: tensor(0.0009) 256 Training Loss: tensor(0.0014) 257 Training Loss: tensor(0.0020) 258 Training Loss: tensor(0.0021) 259 Training Loss: tensor(0.0015) 260 Training Loss: tensor(0.0008) 261 Training Loss: tensor(0.0005) 262 Training Loss: tensor(0.0003) 263 Training Loss: tensor(0.0006) 264 Training Loss: tensor(0.0008) 265 Training Loss: tensor(0.0009) 266 Training Loss: tensor(0.0009) 267 Training Loss: tensor(0.0006) 268 Training Loss: tensor(0.0006) 269 Training Loss: tensor(0.0005) 270 Training Loss: tensor(0.0008) 271 Training Loss: tensor(0.0009) 272 Training Loss: tensor(0.0008) 273 Training Loss: tensor(0.0009) 274 Training Loss: tensor(0.0007) 275 Training Loss: tensor(0.0006) 276 Training Loss: tensor(0.0005) 277 Training Loss: tensor(0.0004) 278 Training Loss: tensor(0.0005) 279 Training Loss: tensor(0.0007) 280 Training Loss: tensor(0.0005) 281 Training Loss: tensor(0.0007) 282 Training Loss: tensor(0.0006) 283 Training Loss: tensor(0.0005) 284 Training Loss: tensor(0.0005) 285 Training Loss: tensor(0.0005) 286 Training Loss: tensor(0.0005) 287 Training Loss: tensor(0.0004) 288 Training Loss: tensor(0.0004) 289 Training Loss: tensor(0.0005) 290 Training Loss: tensor(0.0005) 291 Training Loss: tensor(0.0004) 292 Training Loss: tensor(0.0004) 293 Training Loss: tensor(0.0004) 294 Training Loss: tensor(0.0004) 295 Training Loss: tensor(0.0005) 296 Training Loss: tensor(0.0003) 297 Training Loss: tensor(0.0003) 298 Training Loss: tensor(0.0002) 299 Training Loss: tensor(0.0004) 300 Training Loss: tensor(0.0002) 301 Training Loss: tensor(0.0002) 302 Training Loss: tensor(0.0003) 303 Training Loss: tensor(0.0002) 304 Training Loss: tensor(0.0003) 305 Training Loss: tensor(0.0003) 306 Training Loss: tensor(0.0003) 307 Training Loss: tensor(0.0003) 308 Training Loss: tensor(0.0003) 309 Training Loss: tensor(0.0003) 310 Training Loss: tensor(0.0002) 311 Training Loss: tensor(0.0003) 312 Training Loss: tensor(0.0002) 313 Training Loss: tensor(0.0002) 314 Training Loss: tensor(0.0002) 315 Training Loss: tensor(0.0004) 316 Training Loss: tensor(0.0004) 317 Training Loss: tensor(0.0002) 318 Training Loss: tensor(0.0003) 319 Training Loss: tensor(0.0002) 320 Training Loss: tensor(0.0003) 321 Training Loss: tensor(0.0002) 322 Training Loss: tensor(0.0002) 323 Training Loss: tensor(0.0002) 324 Training Loss: tensor(0.0002) 325 Training Loss: 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355 Training Loss: tensor(0.0007) 356 Training Loss: tensor(0.0009) 357 Training Loss: tensor(0.0013) 358 Training Loss: tensor(0.0017) 359 Training Loss: tensor(0.0018) 360 Training Loss: tensor(0.0021) 361 Training Loss: tensor(0.0020) 362 Training Loss: tensor(0.0020) 363 Training Loss: tensor(0.0018) 364 Training Loss: tensor(0.0010) 365 Training Loss: tensor(0.0006) 366 Training Loss: tensor(0.0002) 367 Training Loss: tensor(0.0007) 368 Training Loss: tensor(0.0007) 369 Training Loss: tensor(0.0012) 370 Training Loss: tensor(0.0016) 371 Training Loss: tensor(0.0013) 372 Training Loss: tensor(0.0013) 373 Training Loss: tensor(0.0007) 374 Training Loss: tensor(0.0007) 375 Training Loss: tensor(0.0005) 376 Training Loss: tensor(0.0003) 377 Training Loss: tensor(0.0005) 378 Training Loss: tensor(0.0002) 379 Training Loss: tensor(0.0003) 380 Training Loss: tensor(0.0003) 381 Training Loss: tensor(0.0005) 382 Training Loss: tensor(0.0006) 383 Training Loss: tensor(0.0006) 384 Training 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In [51]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.4, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.5]$¶
- training with iteration 20000
In [52]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.5, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0006) 1 Training Loss: tensor(0.0013) 2 Training Loss: tensor(0.0016) 3 Training Loss: tensor(0.0023) 4 Training Loss: tensor(0.0018) 5 Training Loss: tensor(0.0016) 6 Training Loss: tensor(0.0007) 7 Training Loss: tensor(0.0013) 8 Training Loss: tensor(0.0015) 9 Training Loss: tensor(0.0018) 10 Training Loss: tensor(0.0016) 11 Training Loss: tensor(0.0005) 12 Training Loss: tensor(0.0017) 13 Training Loss: tensor(0.0018) 14 Training Loss: tensor(0.0018) 15 Training Loss: tensor(0.0014) 16 Training Loss: tensor(0.0009) 17 Training Loss: tensor(0.0022) 18 Training Loss: tensor(0.0021) 19 Training Loss: tensor(0.0018) 20 Training Loss: tensor(0.0003) 21 Training Loss: tensor(0.0013) 22 Training Loss: tensor(0.0016) 23 Training Loss: tensor(0.0014) 24 Training Loss: tensor(0.0006) 25 Training Loss: tensor(0.0008) 26 Training Loss: tensor(0.0017) 27 Training Loss: tensor(0.0019) 28 Training Loss: tensor(0.0010) 29 Training Loss: tensor(0.0004) 30 Training Loss: tensor(0.0013) 31 Training Loss: tensor(0.0022) 32 Training Loss: tensor(0.0016) 33 Training Loss: tensor(0.0006) 34 Training Loss: tensor(0.0009) 35 Training Loss: tensor(0.0019) 36 Training Loss: tensor(0.0011) 37 Training Loss: tensor(0.0007) 38 Training Loss: tensor(0.0006) 39 Training Loss: tensor(0.0011) 40 Training Loss: tensor(0.0007) 41 Training Loss: tensor(0.0004) 42 Training Loss: tensor(0.0004) 43 Training Loss: tensor(0.0005) 44 Training Loss: tensor(0.0004) 45 Training Loss: tensor(0.0005) 46 Training Loss: tensor(0.0002) 47 Training Loss: tensor(0.0005) 48 Training Loss: tensor(0.0003) 49 Training Loss: tensor(0.0005) 50 Training Loss: tensor(0.0005) 51 Training Loss: tensor(0.0009) 52 Training Loss: tensor(0.0004) 53 Training Loss: tensor(0.0004) 54 Training Loss: tensor(0.0004) 55 Training Loss: tensor(0.0008) 56 Training Loss: tensor(0.0004) 57 Training Loss: tensor(0.0004) 58 Training Loss: tensor(0.0003) 59 Training Loss: tensor(0.0007) 60 Training Loss: tensor(0.0006) 61 Training Loss: tensor(0.0004) 62 Training Loss: tensor(0.0003) 63 Training Loss: tensor(0.0003) 64 Training Loss: tensor(0.0003) 65 Training Loss: tensor(0.0003) 66 Training Loss: tensor(0.0002) 67 Training Loss: tensor(0.0002) 68 Training Loss: tensor(0.0002) 69 Training Loss: tensor(0.0002) 70 Training Loss: tensor(0.0002) 71 Training Loss: tensor(0.0002) 72 Training Loss: tensor(0.0002) 73 Training Loss: tensor(0.0002) 74 Training Loss: tensor(0.0002) 75 Training Loss: tensor(0.0001) 76 Training Loss: tensor(0.0002) 77 Training Loss: tensor(0.0001) 78 Training Loss: tensor(0.0001) 79 Training Loss: tensor(0.0001) 80 Training Loss: tensor(0.0001) 81 Training Loss: tensor(8.8643e-05) 82 Training Loss: tensor(7.5406e-05) 83 Training Loss: tensor(9.8718e-05) 84 Training Loss: tensor(8.9846e-05) 85 Training Loss: tensor(0.0001) 86 Training Loss: tensor(0.0001) 87 Training Loss: tensor(0.0002) 88 Training Loss: tensor(0.0002) 89 Training Loss: tensor(0.0002) 90 Training Loss: tensor(0.0001) 91 Training Loss: tensor(0.0002) 92 Training Loss: tensor(0.0001) 93 Training Loss: tensor(0.0001) 94 Training Loss: tensor(0.0001) 95 Training Loss: tensor(0.0001) 96 Training Loss: tensor(0.0002) 97 Training Loss: tensor(0.0001) 98 Training Loss: tensor(0.0002) 99 Training Loss: tensor(0.0001) 100 Training Loss: tensor(0.0001) 101 Training Loss: tensor(0.0002) 102 Training Loss: tensor(0.0001) 103 Training Loss: tensor(0.0001) 104 Training Loss: tensor(0.0001) 105 Training Loss: tensor(0.0001) 106 Training Loss: tensor(0.0002) 107 Training Loss: tensor(0.0001) 108 Training Loss: tensor(0.0001) 109 Training Loss: tensor(0.0002) 110 Training Loss: tensor(0.0003) 111 Training Loss: tensor(0.0002) 112 Training Loss: tensor(0.0001) 113 Training Loss: tensor(0.0001) 114 Training Loss: tensor(0.0002) 115 Training Loss: tensor(0.0002) 116 Training Loss: tensor(0.0001) 117 Training Loss: tensor(0.0001) 118 Training Loss: tensor(0.0002) 119 Training Loss: tensor(0.0002) 120 Training Loss: tensor(0.0001) 121 Training Loss: tensor(0.0001) 122 Training Loss: tensor(0.0002) 123 Training Loss: tensor(0.0002) 124 Training Loss: tensor(0.0001) 125 Training Loss: tensor(0.0001) 126 Training Loss: tensor(0.0002) 127 Training Loss: tensor(9.3500e-05) 128 Training Loss: tensor(0.0002) 129 Training Loss: tensor(0.0001) 130 Training Loss: tensor(0.0001) 131 Training Loss: tensor(0.0001) 132 Training Loss: tensor(0.0002) 133 Training Loss: tensor(9.4917e-05) 134 Training Loss: tensor(0.0001) 135 Training Loss: tensor(0.0001) 136 Training Loss: tensor(0.0001) 137 Training Loss: tensor(0.0001) 138 Training Loss: tensor(8.3929e-05) 139 Training Loss: tensor(0.0001) 140 Training Loss: tensor(0.0001) 141 Training Loss: tensor(0.0001) 142 Training Loss: tensor(0.0001) 143 Training Loss: tensor(8.2715e-05) 144 Training Loss: tensor(9.3239e-05) 145 Training Loss: tensor(8.9124e-05) 146 Training Loss: tensor(8.8444e-05) 147 Training Loss: tensor(6.9661e-05) 148 Training Loss: tensor(8.7897e-05) 149 Training Loss: tensor(0.0001) 150 Training Loss: tensor(0.0001) 151 Training Loss: tensor(0.0001) 152 Training Loss: tensor(9.7463e-05) 153 Training Loss: tensor(0.0001) 154 Training Loss: tensor(9.2575e-05) 155 Training Loss: tensor(9.0012e-05) 156 Training Loss: tensor(8.7372e-05) 157 Training Loss: tensor(9.0160e-05) 158 Training Loss: tensor(8.3446e-05) 159 Training Loss: tensor(9.8223e-05) 160 Training Loss: tensor(0.0001) 161 Training Loss: tensor(8.6780e-05) 162 Training Loss: tensor(8.8548e-05) 163 Training Loss: tensor(9.0850e-05) 164 Training Loss: tensor(0.0001) 165 Training Loss: tensor(0.0001) 166 Training Loss: tensor(0.0002) 167 Training Loss: tensor(0.0001) 168 Training Loss: tensor(9.0033e-05) 169 Training Loss: tensor(0.0001) 170 Training Loss: tensor(0.0001) 171 Training Loss: tensor(0.0001) 172 Training Loss: tensor(8.2271e-05) 173 Training Loss: tensor(8.3804e-05) 174 Training Loss: tensor(8.2488e-05) 175 Training Loss: tensor(9.1149e-05) 176 Training Loss: tensor(9.6080e-05) 177 Training Loss: tensor(0.0001) 178 Training Loss: tensor(0.0001) 179 Training Loss: tensor(0.0002) 180 Training Loss: tensor(0.0001) 181 Training Loss: tensor(0.0001) 182 Training Loss: tensor(0.0001) 183 Training Loss: tensor(0.0001) 184 Training Loss: tensor(0.0002) 185 Training Loss: tensor(7.5607e-05) 186 Training Loss: tensor(0.0002) 187 Training Loss: tensor(9.0906e-05) 188 Training Loss: tensor(0.0002) 189 Training Loss: tensor(8.0333e-05) 190 Training Loss: tensor(0.0001) 191 Training Loss: tensor(7.4569e-05) 192 Training Loss: tensor(0.0001) 193 Training Loss: tensor(0.0001) 194 Training Loss: tensor(0.0001) 195 Training Loss: tensor(0.0002) 196 Training Loss: tensor(0.0001) 197 Training Loss: tensor(0.0001) 198 Training Loss: tensor(8.2200e-05) 199 Training Loss: tensor(0.0001) 200 Training Loss: tensor(0.0002) 201 Training Loss: tensor(0.0001) 202 Training Loss: tensor(0.0002) 203 Training Loss: tensor(0.0002) 204 Training Loss: tensor(0.0001) 205 Training Loss: tensor(0.0002) 206 Training Loss: tensor(0.0003) 207 Training Loss: tensor(0.0002) 208 Training Loss: tensor(0.0002) 209 Training Loss: tensor(0.0001) 210 Training Loss: tensor(0.0002) 211 Training Loss: tensor(0.0001) 212 Training Loss: tensor(0.0002) 213 Training Loss: tensor(0.0001) 214 Training Loss: tensor(0.0001) 215 Training Loss: tensor(0.0001) 216 Training Loss: tensor(0.0001) 217 Training Loss: tensor(0.0001) 218 Training Loss: tensor(0.0001) 219 Training Loss: tensor(9.7720e-05) 220 Training Loss: tensor(0.0001) 221 Training Loss: tensor(0.0001) 222 Training Loss: tensor(0.0001) 223 Training Loss: tensor(0.0001) 224 Training Loss: tensor(8.3913e-05) 225 Training Loss: tensor(0.0001) 226 Training Loss: tensor(0.0001) 227 Training Loss: tensor(0.0001) 228 Training Loss: tensor(0.0001) 229 Training Loss: tensor(9.6947e-05) 230 Training Loss: tensor(0.0001) 231 Training Loss: tensor(9.5453e-05) 232 Training Loss: tensor(0.0002) 233 Training Loss: tensor(0.0002) 234 Training Loss: tensor(0.0002) 235 Training Loss: tensor(0.0001) 236 Training Loss: tensor(0.0002) 237 Training Loss: tensor(0.0001) 238 Training Loss: tensor(0.0002) 239 Training Loss: tensor(0.0002) 240 Training Loss: tensor(0.0001) 241 Training Loss: tensor(0.0002) 242 Training Loss: tensor(0.0001) 243 Training Loss: tensor(0.0001) 244 Training Loss: tensor(0.0002) 245 Training Loss: tensor(0.0002) 246 Training Loss: tensor(0.0001) 247 Training Loss: tensor(0.0001) 248 Training Loss: tensor(0.0001) 249 Training Loss: tensor(0.0001) 250 Training Loss: tensor(0.0002) 251 Training Loss: tensor(0.0002) 252 Training Loss: tensor(0.0001) 253 Training Loss: tensor(8.5148e-05) 254 Training Loss: tensor(0.0002) 255 Training Loss: tensor(0.0002) 256 Training Loss: tensor(0.0003) 257 Training Loss: tensor(0.0003) 258 Training Loss: tensor(0.0002) 259 Training Loss: tensor(7.6261e-05) 260 Training Loss: tensor(0.0002) 261 Training Loss: tensor(0.0003) 262 Training Loss: tensor(0.0003) 263 Training Loss: tensor(0.0003) 264 Training Loss: tensor(0.0001) 265 Training Loss: tensor(0.0001) 266 Training Loss: tensor(9.0924e-05) 267 Training Loss: tensor(0.0002) 268 Training Loss: tensor(0.0002) 269 Training Loss: tensor(0.0001) 270 Training Loss: tensor(0.0001) 271 Training Loss: tensor(0.0001) 272 Training Loss: tensor(0.0002) 273 Training Loss: tensor(0.0003) 274 Training Loss: tensor(0.0003) 275 Training Loss: tensor(0.0002) 276 Training Loss: tensor(0.0002) 277 Training Loss: tensor(9.2284e-05) 278 Training Loss: tensor(0.0002) 279 Training Loss: tensor(0.0002) 280 Training Loss: tensor(0.0002) 281 Training Loss: tensor(0.0002) 282 Training Loss: tensor(0.0001) 283 Training Loss: tensor(0.0001) 284 Training Loss: tensor(0.0001) 285 Training Loss: tensor(0.0002) 286 Training Loss: tensor(0.0001) 287 Training Loss: tensor(0.0001) 288 Training Loss: tensor(9.0642e-05) 289 Training Loss: tensor(0.0002) 290 Training Loss: tensor(0.0002) 291 Training Loss: tensor(0.0003) 292 Training Loss: tensor(0.0003) 293 Training Loss: tensor(0.0002) 294 Training Loss: tensor(0.0001) 295 Training Loss: tensor(0.0001) 296 Training Loss: tensor(0.0002) 297 Training Loss: tensor(0.0001) 298 Training Loss: tensor(0.0002) 299 Training Loss: tensor(0.0002) 300 Training Loss: tensor(0.0002) 301 Training Loss: tensor(9.2239e-05) 302 Training Loss: tensor(0.0001) 303 Training Loss: tensor(0.0001) 304 Training Loss: tensor(0.0001) 305 Training Loss: tensor(0.0002) 306 Training Loss: tensor(0.0002) 307 Training Loss: tensor(0.0002) 308 Training Loss: tensor(0.0004) 309 Training Loss: tensor(0.0004) 310 Training Loss: tensor(0.0004) 311 Training Loss: tensor(0.0003) 312 Training Loss: tensor(0.0001) 313 Training Loss: tensor(0.0002) 314 Training Loss: tensor(0.0004) 315 Training Loss: tensor(0.0010) 316 Training Loss: tensor(0.0012) 317 Training Loss: tensor(0.0011) 318 Training Loss: tensor(0.0007) 319 Training Loss: tensor(0.0003) 320 Training Loss: tensor(0.0001) 321 Training Loss: tensor(0.0003) 322 Training Loss: tensor(0.0007) 323 Training Loss: tensor(0.0009) 324 Training Loss: tensor(0.0009) 325 Training Loss: tensor(0.0004) 326 Training Loss: tensor(0.0003) 327 Training Loss: tensor(0.0001) 328 Training Loss: tensor(0.0003) 329 Training Loss: tensor(0.0004) 330 Training Loss: tensor(0.0005) 331 Training Loss: tensor(0.0004) 332 Training Loss: tensor(0.0003) 333 Training Loss: tensor(0.0001) 334 Training Loss: tensor(0.0002) 335 Training Loss: tensor(0.0002) 336 Training Loss: tensor(0.0004) 337 Training Loss: tensor(0.0004) 338 Training Loss: tensor(0.0002) 339 Training Loss: tensor(0.0002) 340 Training Loss: tensor(0.0001) 341 Training Loss: tensor(0.0002) 342 Training Loss: tensor(0.0003) 343 Training Loss: tensor(0.0003) 344 Training Loss: tensor(0.0002) 345 Training Loss: tensor(9.4748e-05) 346 Training Loss: tensor(0.0002) 347 Training Loss: tensor(0.0003) 348 Training Loss: tensor(0.0003) 349 Training Loss: tensor(0.0004) 350 Training Loss: tensor(0.0002) 351 Training Loss: tensor(0.0002) 352 Training Loss: tensor(8.9142e-05) 353 Training Loss: tensor(0.0002) 354 Training Loss: tensor(0.0002) 355 Training Loss: tensor(0.0002) 356 Training Loss: tensor(0.0002) 357 Training Loss: tensor(0.0002) 358 Training Loss: tensor(0.0002) 359 Training Loss: tensor(0.0002) 360 Training Loss: tensor(0.0002) 361 Training Loss: tensor(0.0002) 362 Training Loss: tensor(0.0004) 363 Training Loss: tensor(0.0003) 364 Training Loss: tensor(0.0003) 365 Training Loss: tensor(0.0002) 366 Training Loss: tensor(0.0003) 367 Training Loss: tensor(0.0002) 368 Training Loss: tensor(0.0002) 369 Training Loss: tensor(0.0002) 370 Training Loss: tensor(0.0001) 371 Training Loss: tensor(0.0001) 372 Training Loss: tensor(0.0002) 373 Training Loss: tensor(0.0001) 374 Training Loss: tensor(0.0002) 375 Training Loss: tensor(8.6829e-05) 376 Training Loss: tensor(0.0001) 377 Training Loss: tensor(0.0001) 378 Training Loss: tensor(0.0002) 379 Training Loss: tensor(0.0002) 380 Training Loss: tensor(0.0002) 381 Training Loss: tensor(0.0002) 382 Training Loss: tensor(0.0002) 383 Training Loss: tensor(0.0002) 384 Training Loss: tensor(0.0002) 385 Training Loss: tensor(0.0002) 386 Training Loss: tensor(0.0003) 387 Training Loss: tensor(0.0002) 388 Training Loss: tensor(0.0002) 389 Training Loss: tensor(0.0001) 390 Training Loss: tensor(0.0001) 391 Training Loss: tensor(0.0002) 392 Training Loss: tensor(0.0002) 393 Training Loss: tensor(0.0003) 394 Training Loss: tensor(0.0002) 395 Training Loss: tensor(0.0002) 396 Training Loss: tensor(0.0002) 397 Training Loss: tensor(0.0002) 398 Training Loss: tensor(0.0002) 399 Training Loss: tensor(0.0003) 400 Training Loss: tensor(0.0004) 401 Training Loss: tensor(0.0004) 402 Training Loss: tensor(0.0004) 403 Training Loss: tensor(0.0003) 404 Training Loss: tensor(0.0003) 405 Training Loss: tensor(0.0003) 406 Training Loss: tensor(0.0004) 407 Training Loss: tensor(0.0007) 408 Training Loss: tensor(0.0007) 409 Training Loss: tensor(0.0008) 410 Training Loss: tensor(0.0011) 411 Training Loss: tensor(0.0009) 412 Training Loss: tensor(0.0012) 413 Training Loss: tensor(0.0006) 414 Training Loss: tensor(0.0014) 415 Training Loss: tensor(0.0008) 416 Training Loss: tensor(0.0008) 417 Training Loss: tensor(0.0010) 418 Training Loss: tensor(0.0006) 419 Training Loss: tensor(0.0009) 420 Training Loss: tensor(0.0004) 421 Training Loss: tensor(0.0007) 422 Training Loss: tensor(0.0004) 423 Training Loss: tensor(0.0009) 424 Training Loss: tensor(0.0007) 425 Training Loss: tensor(0.0010) 426 Training Loss: tensor(0.0005) 427 Training Loss: tensor(0.0012) 428 Training Loss: tensor(0.0004) 429 Training Loss: tensor(0.0011) 430 Training Loss: tensor(0.0005) 431 Training Loss: tensor(0.0013) 432 Training Loss: tensor(0.0013) 433 Training Loss: tensor(0.0009) 434 Training Loss: tensor(0.0014) 435 Training Loss: tensor(0.0006) 436 Training Loss: tensor(0.0015) 437 Training Loss: tensor(0.0007) 438 Training Loss: tensor(0.0016) 439 Training Loss: tensor(0.0016) 440 Training Loss: tensor(0.0015) 441 Training Loss: tensor(0.0009) 442 Training Loss: tensor(0.0006) 443 Training Loss: tensor(0.0007) 444 Training Loss: tensor(0.0003) 445 Training Loss: tensor(0.0005) 446 Training Loss: tensor(0.0004) 447 Training Loss: tensor(0.0004) 448 Training Loss: tensor(0.0003) 449 Training Loss: tensor(0.0003) 450 Training Loss: tensor(0.0002) 451 Training Loss: tensor(0.0003) 452 Training Loss: tensor(0.0003) 453 Training Loss: tensor(0.0003) 454 Training Loss: tensor(0.0003) 455 Training Loss: tensor(0.0004) 456 Training Loss: tensor(0.0003) 457 Training Loss: tensor(0.0005) 458 Training Loss: tensor(0.0006) 459 Training Loss: tensor(0.0004) 460 Training Loss: tensor(0.0004) 461 Training Loss: tensor(0.0003) 462 Training Loss: tensor(0.0002) 463 Training Loss: tensor(0.0004) 464 Training Loss: tensor(0.0002) 465 Training Loss: tensor(0.0003) 466 Training Loss: tensor(0.0003) 467 Training Loss: tensor(0.0002) 468 Training Loss: tensor(0.0002) 469 Training Loss: tensor(0.0002) 470 Training Loss: tensor(0.0003) 471 Training Loss: tensor(0.0001) 472 Training Loss: tensor(0.0003) 473 Training Loss: tensor(0.0001) 474 Training Loss: tensor(0.0003) 475 Training Loss: tensor(0.0002) 476 Training Loss: tensor(0.0002) 477 Training Loss: tensor(0.0001) 478 Training Loss: tensor(0.0002) 479 Training Loss: tensor(0.0002) 480 Training Loss: tensor(0.0002) 481 Training Loss: tensor(0.0004) 482 Training Loss: tensor(0.0004) 483 Training Loss: tensor(0.0005) 484 Training Loss: tensor(0.0003) 485 Training Loss: tensor(0.0003) 486 Training Loss: tensor(7.9969e-05) 487 Training Loss: tensor(0.0003) 488 Training Loss: tensor(0.0003) 489 Training Loss: tensor(0.0004) 490 Training Loss: tensor(0.0004) 491 Training Loss: tensor(0.0004) 492 Training Loss: tensor(0.0002) 493 Training Loss: tensor(0.0002) 494 Training Loss: tensor(0.0004) 495 Training Loss: tensor(0.0009) 496 Training Loss: tensor(0.0014) 497 Training Loss: tensor(0.0016) 498 Training Loss: tensor(0.0014) 499 Training Loss: tensor(0.0006) 500 Training Loss: tensor(0.0003) 501 Training Loss: tensor(0.0003) 502 Training Loss: tensor(0.0009) 503 Training Loss: tensor(0.0018) 504 Training Loss: tensor(0.0024) 505 Training Loss: tensor(0.0024) 506 Training Loss: tensor(0.0017) 507 Training Loss: tensor(0.0007) 508 Training Loss: tensor(0.0003) 509 Training Loss: tensor(0.0007) 510 Training Loss: tensor(0.0013) 511 Training Loss: tensor(0.0014) 512 Training Loss: tensor(0.0007) 513 Training Loss: tensor(0.0009) 514 Training Loss: tensor(0.0004) 515 Training Loss: tensor(0.0018) 516 Training Loss: tensor(0.0016) 517 Training Loss: tensor(0.0016) 518 Training Loss: tensor(0.0007) 519 Training Loss: tensor(0.0009) 520 Training Loss: tensor(0.0013) 521 Training Loss: tensor(0.0013) 522 Training Loss: tensor(0.0010) 523 Training Loss: tensor(0.0005) 524 Training Loss: tensor(0.0006) 525 Training Loss: tensor(0.0006) 526 Training Loss: tensor(0.0006) 527 Training Loss: tensor(0.0005) 528 Training Loss: tensor(0.0002) 529 Training Loss: tensor(0.0008) 530 Training Loss: tensor(0.0008) 531 Training Loss: tensor(0.0005) 532 Training Loss: tensor(0.0006) 533 Training Loss: tensor(0.0003) 534 Training Loss: tensor(0.0005) 535 Training Loss: tensor(0.0005) 536 Training Loss: tensor(0.0004) 537 Training Loss: tensor(0.0005) 538 Training Loss: tensor(0.0004) 539 Training Loss: tensor(0.0005) 540 Training Loss: tensor(0.0002) 541 Training Loss: tensor(0.0004) 542 Training Loss: tensor(0.0004) 543 Training Loss: tensor(0.0004) 544 Training Loss: tensor(0.0003) 545 Training Loss: tensor(0.0003) 546 Training Loss: tensor(0.0003) 547 Training Loss: tensor(0.0001) 548 Training Loss: tensor(0.0003) 549 Training Loss: tensor(0.0004) 550 Training Loss: tensor(0.0004) 551 Training Loss: tensor(0.0003) 552 Training Loss: tensor(0.0002) 553 Training Loss: tensor(0.0003) 554 Training Loss: tensor(0.0006) 555 Training Loss: tensor(0.0008) 556 Training Loss: tensor(0.0008) 557 Training Loss: tensor(0.0003) 558 Training Loss: tensor(0.0003) 559 Training Loss: tensor(0.0003) 560 Training Loss: tensor(0.0007) 561 Training Loss: tensor(0.0008) 562 Training Loss: tensor(0.0007) 563 Training Loss: tensor(0.0002) 564 Training Loss: tensor(0.0005) 565 Training Loss: tensor(0.0004) 566 Training Loss: tensor(0.0006) 567 Training Loss: tensor(0.0003) 568 Training Loss: tensor(0.0002) 569 Training Loss: tensor(0.0003) 570 Training Loss: tensor(0.0005) 571 Training Loss: tensor(0.0003) 572 Training Loss: tensor(0.0003) 573 Training Loss: tensor(0.0002) 574 Training Loss: tensor(0.0004) 575 Training Loss: tensor(0.0004) 576 Training Loss: tensor(0.0004) 577 Training Loss: tensor(0.0002) 578 Training Loss: tensor(0.0003) 579 Training Loss: tensor(0.0002) 580 Training Loss: tensor(0.0006) 581 Training Loss: tensor(0.0002) 582 Training Loss: tensor(0.0003) 583 Training Loss: tensor(0.0002) 584 Training Loss: tensor(0.0002) 585 Training Loss: tensor(0.0002) 586 Training Loss: tensor(0.0003) 587 Training Loss: tensor(0.0003) 588 Training Loss: tensor(0.0003) 589 Training Loss: tensor(0.0003) 590 Training Loss: tensor(0.0002) 591 Training Loss: tensor(0.0004) 592 Training Loss: tensor(0.0001) 593 Training Loss: tensor(0.0003) 594 Training Loss: tensor(0.0002) 595 Training Loss: tensor(0.0002) 596 Training Loss: tensor(0.0001) 597 Training Loss: tensor(0.0003) 598 Training Loss: tensor(0.0002) 599 Training Loss: tensor(0.0002) 600 Training Loss: tensor(0.0002) 601 Training Loss: tensor(0.0002) 602 Training Loss: tensor(0.0001) 603 Training Loss: tensor(0.0002) 604 Training Loss: tensor(0.0002) 605 Training Loss: tensor(0.0003) 606 Training Loss: tensor(0.0002) 607 Training Loss: tensor(0.0001) 608 Training Loss: tensor(0.0002) 609 Training Loss: tensor(0.0005) 610 Training Loss: tensor(0.0005) 611 Training Loss: tensor(0.0004) 612 Training Loss: tensor(0.0003) 613 Training Loss: tensor(0.0003) 614 Training Loss: tensor(0.0002) 615 Training Loss: tensor(0.0002) 616 Training Loss: tensor(0.0001) 617 Training Loss: tensor(0.0004) 618 Training Loss: tensor(0.0005) 619 Training Loss: tensor(0.0004) 620 Training Loss: tensor(0.0003) 621 Training Loss: tensor(0.0001) 622 Training Loss: tensor(0.0002) 623 Training Loss: tensor(0.0003) 624 Training Loss: tensor(0.0003) 625 Training Loss: tensor(0.0004) 626 Training Loss: tensor(0.0001) 627 Training Loss: tensor(0.0001) 628 Training Loss: tensor(0.0002) 629 Training Loss: tensor(0.0003) 630 Training Loss: tensor(0.0004) 631 Training Loss: tensor(0.0003) 632 Training Loss: tensor(0.0003) 633 Training Loss: tensor(0.0002) 634 Training Loss: tensor(0.0002) 635 Training Loss: tensor(0.0003) 636 Training Loss: tensor(0.0003) 637 Training Loss: tensor(0.0003) 638 Training Loss: tensor(0.0002) 639 Training Loss: tensor(0.0002) 640 Training Loss: tensor(0.0001) 641 Training Loss: tensor(0.0001) 642 Training Loss: tensor(0.0002) 643 Training Loss: tensor(0.0002) 644 Training Loss: tensor(0.0004) 645 Training Loss: tensor(0.0003) 646 Training Loss: tensor(0.0003) 647 Training Loss: tensor(0.0001) 648 Training Loss: tensor(0.0002) 649 Training Loss: tensor(0.0002) 650 Training Loss: tensor(0.0003) 651 Training Loss: tensor(0.0003) 652 Training Loss: tensor(0.0002) 653 Training Loss: tensor(0.0001) 654 Training Loss: tensor(0.0001) 655 Training Loss: tensor(0.0002) 656 Training Loss: tensor(0.0003) 657 Training Loss: tensor(0.0003) 658 Training Loss: tensor(9.0991e-05) 659 Training Loss: tensor(0.0002) 660 Training Loss: tensor(0.0002) 661 Training Loss: tensor(0.0004) 662 Training Loss: tensor(0.0004) 663 Training Loss: tensor(0.0003) 664 Training Loss: tensor(0.0001) 665 Training Loss: tensor(0.0002) 666 Training Loss: tensor(0.0001) 667 Training Loss: tensor(0.0002) 668 Training Loss: tensor(0.0002) 669 Training Loss: tensor(0.0001) 670 Training Loss: tensor(0.0002) 671 Training Loss: tensor(0.0002) 672 Training Loss: tensor(0.0002) 673 Training Loss: tensor(0.0001) 674 Training Loss: tensor(0.0001) 675 Training Loss: tensor(0.0002) 676 Training Loss: tensor(0.0002) 677 Training Loss: tensor(0.0001) 678 Training Loss: tensor(0.0002) 679 Training Loss: tensor(0.0001) 680 Training Loss: tensor(0.0002) 681 Training Loss: tensor(0.0001) 682 Training Loss: tensor(0.0003) 683 Training Loss: tensor(0.0003) 684 Training Loss: tensor(0.0004) 685 Training Loss: tensor(0.0002) 686 Training Loss: tensor(0.0003) 687 Training Loss: tensor(0.0002) 688 Training Loss: tensor(0.0003) 689 Training Loss: tensor(0.0003) 690 Training Loss: tensor(0.0003) 691 Training Loss: tensor(0.0003) 692 Training Loss: tensor(0.0002) 693 Training Loss: tensor(0.0002) 694 Training Loss: tensor(9.7073e-05) 695 Training Loss: tensor(0.0002) 696 Training Loss: tensor(0.0002) 697 Training Loss: tensor(0.0004) 698 Training Loss: tensor(0.0003) 699 Training Loss: tensor(0.0002) 700 Training Loss: 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Training Loss: tensor(0.0001) 19840 Training Loss: tensor(0.0002) 19841 Training Loss: tensor(0.0002) 19842 Training Loss: tensor(0.0002) 19843 Training Loss: tensor(0.0001) 19844 Training Loss: tensor(0.0001) 19845 Training Loss: tensor(0.0002) 19846 Training Loss: tensor(0.0002) 19847 Training Loss: tensor(0.0002) 19848 Training Loss: tensor(0.0001) 19849 Training Loss: tensor(7.8721e-05) 19850 Training Loss: tensor(7.7079e-05) 19851 Training Loss: tensor(0.0001) 19852 Training Loss: tensor(7.0882e-05) 19853 Training Loss: tensor(0.0001) 19854 Training Loss: tensor(7.7653e-05) 19855 Training Loss: tensor(5.3124e-05) 19856 Training Loss: tensor(4.7744e-05) 19857 Training Loss: tensor(7.1019e-05) 19858 Training Loss: tensor(8.1216e-05) 19859 Training Loss: tensor(9.1256e-05) 19860 Training Loss: tensor(8.0988e-05) 19861 Training Loss: tensor(4.1519e-05) 19862 Training Loss: tensor(5.8434e-05) 19863 Training Loss: tensor(7.8663e-05) 19864 Training Loss: tensor(0.0001) 19865 Training Loss: tensor(0.0002) 19866 Training Loss: tensor(0.0002) 19867 Training Loss: tensor(0.0002) 19868 Training Loss: tensor(6.4472e-05) 19869 Training Loss: tensor(4.1172e-05) 19870 Training Loss: tensor(0.0001) 19871 Training Loss: tensor(0.0003) 19872 Training Loss: tensor(0.0003) 19873 Training Loss: tensor(0.0003) 19874 Training Loss: tensor(9.9780e-05) 19875 Training Loss: tensor(5.3726e-05) 19876 Training Loss: tensor(0.0001) 19877 Training Loss: tensor(0.0002) 19878 Training Loss: tensor(0.0003) 19879 Training Loss: tensor(0.0004) 19880 Training Loss: tensor(0.0004) 19881 Training Loss: tensor(0.0003) 19882 Training Loss: tensor(0.0002) 19883 Training Loss: tensor(5.2460e-05) 19884 Training Loss: tensor(0.0002) 19885 Training Loss: tensor(0.0002) 19886 Training Loss: tensor(0.0003) 19887 Training Loss: tensor(0.0002) 19888 Training Loss: tensor(0.0002) 19889 Training Loss: tensor(0.0001) 19890 Training Loss: tensor(8.5222e-05) 19891 Training Loss: tensor(0.0002) 19892 Training Loss: tensor(0.0002) 19893 Training Loss: tensor(0.0003) 19894 Training Loss: tensor(0.0001) 19895 Training Loss: tensor(0.0002) 19896 Training Loss: tensor(0.0003) 19897 Training Loss: tensor(0.0005) 19898 Training Loss: tensor(0.0004) 19899 Training Loss: tensor(0.0003) 19900 Training Loss: tensor(0.0002) 19901 Training Loss: tensor(0.0001) 19902 Training Loss: tensor(0.0001) 19903 Training Loss: tensor(0.0002) 19904 Training Loss: tensor(0.0001) 19905 Training Loss: tensor(0.0001) 19906 Training Loss: tensor(9.8811e-05) 19907 Training Loss: tensor(0.0001) 19908 Training Loss: tensor(0.0001) 19909 Training Loss: tensor(0.0002) 19910 Training Loss: tensor(0.0002) 19911 Training Loss: tensor(0.0001) 19912 Training Loss: tensor(0.0002) 19913 Training Loss: tensor(6.2401e-05) 19914 Training Loss: tensor(0.0001) 19915 Training Loss: tensor(9.0300e-05) 19916 Training Loss: tensor(0.0001) 19917 Training Loss: tensor(0.0001) 19918 Training Loss: tensor(0.0001) 19919 Training Loss: tensor(0.0001) 19920 Training Loss: tensor(0.0001) 19921 Training Loss: tensor(0.0001) 19922 Training Loss: tensor(0.0002) 19923 Training Loss: tensor(0.0002) 19924 Training Loss: tensor(0.0001) 19925 Training Loss: tensor(0.0001) 19926 Training Loss: tensor(0.0001) 19927 Training Loss: tensor(0.0001) 19928 Training Loss: tensor(0.0001) 19929 Training Loss: tensor(0.0001) 19930 Training Loss: tensor(7.7061e-05) 19931 Training Loss: tensor(0.0002) 19932 Training Loss: tensor(0.0001) 19933 Training Loss: tensor(0.0002) 19934 Training Loss: tensor(0.0001) 19935 Training Loss: tensor(0.0001) 19936 Training Loss: tensor(0.0003) 19937 Training Loss: tensor(0.0001) 19938 Training Loss: tensor(0.0003) 19939 Training Loss: tensor(0.0003) 19940 Training Loss: tensor(0.0004) 19941 Training Loss: tensor(0.0004) 19942 Training Loss: tensor(0.0003) 19943 Training Loss: tensor(0.0003) 19944 Training Loss: tensor(0.0002) 19945 Training Loss: tensor(0.0002) 19946 Training Loss: tensor(0.0001) 19947 Training Loss: tensor(9.5364e-05) 19948 Training Loss: tensor(0.0001) 19949 Training Loss: tensor(8.7057e-05) 19950 Training Loss: tensor(0.0001) 19951 Training Loss: tensor(0.0002) 19952 Training Loss: tensor(0.0002) 19953 Training Loss: tensor(6.5188e-05) 19954 Training Loss: tensor(0.0002) 19955 Training Loss: tensor(0.0002) 19956 Training Loss: tensor(0.0005) 19957 Training Loss: tensor(0.0007) 19958 Training Loss: tensor(0.0015) 19959 Training Loss: tensor(0.0023) 19960 Training Loss: tensor(0.0020) 19961 Training Loss: tensor(0.0012) 19962 Training Loss: tensor(0.0004) 19963 Training Loss: tensor(0.0004) 19964 Training Loss: tensor(0.0013) 19965 Training Loss: tensor(0.0018) 19966 Training Loss: tensor(0.0012) 19967 Training Loss: tensor(0.0004) 19968 Training Loss: tensor(0.0004) 19969 Training Loss: tensor(0.0005) 19970 Training Loss: tensor(0.0007) 19971 Training Loss: tensor(0.0007) 19972 Training Loss: tensor(0.0006) 19973 Training Loss: tensor(0.0005) 19974 Training Loss: tensor(0.0003) 19975 Training Loss: tensor(0.0003) 19976 Training Loss: tensor(0.0004) 19977 Training Loss: tensor(0.0003) 19978 Training Loss: tensor(0.0005) 19979 Training Loss: tensor(0.0003) 19980 Training Loss: tensor(0.0002) 19981 Training Loss: tensor(0.0004) 19982 Training Loss: tensor(0.0005) 19983 Training Loss: tensor(0.0006) 19984 Training Loss: tensor(0.0004) 19985 Training Loss: tensor(0.0002) 19986 Training Loss: tensor(0.0003) 19987 Training Loss: tensor(0.0005) 19988 Training Loss: tensor(0.0007) 19989 Training Loss: tensor(0.0004) 19990 Training Loss: tensor(0.0006) 19991 Training Loss: tensor(0.0004) 19992 Training Loss: tensor(0.0005) 19993 Training Loss: tensor(0.0003) 19994 Training Loss: tensor(0.0002) 19995 Training Loss: tensor(0.0003) 19996 Training Loss: tensor(0.0004) 19997 Training Loss: tensor(0.0003) 19998 Training Loss: tensor(0.0003) 19999 Training Loss: tensor(0.0003)
In [53]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.5, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.6]$¶
- training with iteration 20000
In [54]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.6, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0002) 1 Training Loss: tensor(0.0001) 2 Training Loss: tensor(0.0002) 3 Training Loss: tensor(0.0001) 4 Training Loss: tensor(0.0003) 5 Training Loss: tensor(0.0002) 6 Training Loss: tensor(0.0001) 7 Training Loss: tensor(0.0001) 8 Training Loss: tensor(0.0002) 9 Training Loss: tensor(0.0001) 10 Training Loss: tensor(0.0002) 11 Training Loss: tensor(8.2040e-05) 12 Training Loss: tensor(0.0002) 13 Training Loss: tensor(8.9003e-05) 14 Training Loss: tensor(0.0001) 15 Training Loss: tensor(0.0001) 16 Training Loss: tensor(9.5132e-05) 17 Training Loss: tensor(0.0001) 18 Training Loss: tensor(0.0001) 19 Training Loss: tensor(6.3360e-05) 20 Training Loss: tensor(0.0001) 21 Training Loss: tensor(0.0001) 22 Training Loss: tensor(0.0002) 23 Training Loss: tensor(0.0002) 24 Training Loss: tensor(0.0001) 25 Training Loss: tensor(0.0002) 26 Training Loss: tensor(6.4344e-05) 27 Training Loss: tensor(0.0001) 28 Training Loss: tensor(9.9837e-05) 29 Training Loss: tensor(9.8404e-05) 30 Training Loss: tensor(9.4901e-05) 31 Training Loss: tensor(0.0001) 32 Training Loss: tensor(8.0098e-05) 33 Training Loss: tensor(0.0001) 34 Training Loss: tensor(9.2765e-05) 35 Training Loss: tensor(0.0001) 36 Training Loss: tensor(6.7187e-05) 37 Training Loss: tensor(9.3972e-05) 38 Training Loss: tensor(6.1895e-05) 39 Training Loss: tensor(6.1282e-05) 40 Training Loss: tensor(9.1583e-05) 41 Training Loss: tensor(8.2070e-05) 42 Training Loss: tensor(9.4067e-05) 43 Training Loss: tensor(0.0001) 44 Training Loss: tensor(8.6521e-05) 45 Training Loss: tensor(0.0001) 46 Training Loss: tensor(0.0001) 47 Training Loss: tensor(0.0002) 48 Training Loss: tensor(8.8214e-05) 49 Training Loss: tensor(0.0001) 50 Training Loss: tensor(0.0001) 51 Training Loss: tensor(0.0001) 52 Training Loss: tensor(6.7562e-05) 53 Training Loss: tensor(8.3156e-05) 54 Training Loss: tensor(7.7602e-05) 55 Training Loss: tensor(7.9807e-05) 56 Training Loss: tensor(5.2331e-05) 57 Training Loss: tensor(9.4121e-05) 58 Training Loss: tensor(3.7292e-05) 59 Training Loss: tensor(7.0162e-05) 60 Training Loss: tensor(5.4661e-05) 61 Training Loss: tensor(4.7899e-05) 62 Training Loss: tensor(6.1421e-05) 63 Training Loss: tensor(4.0150e-05) 64 Training Loss: tensor(3.3668e-05) 65 Training Loss: tensor(5.0575e-05) 66 Training Loss: tensor(3.8337e-05) 67 Training Loss: tensor(4.2814e-05) 68 Training Loss: tensor(4.8259e-05) 69 Training Loss: tensor(4.5888e-05) 70 Training Loss: tensor(5.2898e-05) 71 Training Loss: tensor(4.8021e-05) 72 Training Loss: tensor(5.2330e-05) 73 Training Loss: tensor(6.3191e-05) 74 Training Loss: tensor(6.0811e-05) 75 Training Loss: tensor(9.9222e-05) 76 Training Loss: tensor(0.0001) 77 Training Loss: tensor(0.0001) 78 Training Loss: tensor(7.8698e-05) 79 Training Loss: tensor(0.0001) 80 Training Loss: tensor(8.6886e-05) 81 Training Loss: tensor(0.0001) 82 Training Loss: tensor(9.4298e-05) 83 Training Loss: tensor(6.1543e-05) 84 Training Loss: tensor(8.7132e-05) 85 Training Loss: tensor(6.9118e-05) 86 Training Loss: tensor(0.0001) 87 Training Loss: tensor(5.9602e-05) 88 Training Loss: tensor(6.6949e-05) 89 Training Loss: tensor(5.9720e-05) 90 Training Loss: tensor(7.6688e-05) 91 Training Loss: tensor(8.8253e-05) 92 Training Loss: tensor(7.6847e-05) 93 Training Loss: tensor(9.1997e-05) 94 Training Loss: tensor(0.0001) 95 Training Loss: tensor(6.8143e-05) 96 Training Loss: tensor(9.1370e-05) 97 Training Loss: tensor(9.8799e-05) 98 Training Loss: tensor(0.0001) 99 Training Loss: tensor(0.0002) 100 Training Loss: tensor(0.0002) 101 Training Loss: tensor(0.0001) 102 Training Loss: tensor(7.0625e-05) 103 Training Loss: tensor(7.7803e-05) 104 Training Loss: tensor(6.6768e-05) 105 Training Loss: tensor(8.8114e-05) 106 Training Loss: tensor(9.8583e-05) 107 Training Loss: tensor(9.9431e-05) 108 Training Loss: tensor(8.7713e-05) 109 Training Loss: tensor(5.6057e-05) 110 Training Loss: tensor(5.9402e-05) 111 Training Loss: tensor(7.6898e-05) 112 Training Loss: tensor(5.7054e-05) 113 Training Loss: tensor(6.8066e-05) 114 Training Loss: tensor(6.3453e-05) 115 Training Loss: tensor(5.9827e-05) 116 Training Loss: tensor(5.4678e-05) 117 Training Loss: tensor(4.8936e-05) 118 Training Loss: tensor(3.8734e-05) 119 Training Loss: tensor(5.5513e-05) 120 Training Loss: tensor(6.5857e-05) 121 Training Loss: tensor(7.7129e-05) 122 Training Loss: tensor(8.2726e-05) 123 Training Loss: tensor(9.2864e-05) 124 Training Loss: tensor(6.6243e-05) 125 Training Loss: tensor(5.7768e-05) 126 Training Loss: tensor(3.4293e-05) 127 Training Loss: tensor(5.2359e-05) 128 Training Loss: tensor(6.3287e-05) 129 Training Loss: tensor(4.7176e-05) 130 Training Loss: tensor(5.3623e-05) 131 Training Loss: tensor(5.0591e-05) 132 Training Loss: tensor(3.6033e-05) 133 Training Loss: tensor(4.2767e-05) 134 Training Loss: tensor(6.1133e-05) 135 Training Loss: tensor(8.8156e-05) 136 Training Loss: tensor(9.5241e-05) 137 Training Loss: tensor(7.7191e-05) 138 Training Loss: tensor(7.3597e-05) 139 Training Loss: tensor(5.2649e-05) 140 Training Loss: tensor(4.6594e-05) 141 Training Loss: tensor(3.3411e-05) 142 Training Loss: tensor(3.9244e-05) 143 Training Loss: tensor(4.3488e-05) 144 Training Loss: tensor(5.4008e-05) 145 Training Loss: tensor(3.6992e-05) 146 Training Loss: tensor(3.6679e-05) 147 Training Loss: tensor(3.3073e-05) 148 Training Loss: tensor(4.0380e-05) 149 Training Loss: tensor(5.3575e-05) 150 Training Loss: tensor(6.4587e-05) 151 Training Loss: tensor(8.2908e-05) 152 Training Loss: tensor(8.3534e-05) 153 Training Loss: tensor(0.0001) 154 Training Loss: tensor(6.7992e-05) 155 Training Loss: tensor(0.0002) 156 Training Loss: tensor(7.6698e-05) 157 Training Loss: tensor(0.0001) 158 Training Loss: tensor(7.0962e-05) 159 Training Loss: tensor(0.0001) 160 Training Loss: tensor(6.6048e-05) 161 Training Loss: tensor(0.0001) 162 Training Loss: tensor(0.0001) 163 Training Loss: tensor(0.0002) 164 Training Loss: tensor(0.0001) 165 Training Loss: tensor(0.0001) 166 Training Loss: tensor(0.0001) 167 Training Loss: tensor(0.0001) 168 Training Loss: tensor(9.0572e-05) 169 Training Loss: tensor(9.2829e-05) 170 Training Loss: tensor(0.0002) 171 Training Loss: tensor(0.0002) 172 Training Loss: tensor(0.0002) 173 Training Loss: tensor(0.0002) 174 Training Loss: tensor(0.0001) 175 Training Loss: tensor(0.0002) 176 Training Loss: tensor(5.5542e-05) 177 Training Loss: tensor(0.0002) 178 Training Loss: tensor(0.0002) 179 Training Loss: tensor(0.0002) 180 Training Loss: tensor(0.0001) 181 Training Loss: tensor(8.6086e-05) 182 Training Loss: tensor(0.0001) 183 Training Loss: tensor(9.6021e-05) 184 Training Loss: tensor(0.0002) 185 Training Loss: tensor(0.0002) 186 Training Loss: tensor(0.0003) 187 Training Loss: tensor(0.0003) 188 Training Loss: tensor(0.0003) 189 Training Loss: tensor(0.0001) 190 Training Loss: tensor(0.0004) 191 Training Loss: tensor(8.6955e-05) 192 Training Loss: tensor(0.0005) 193 Training Loss: tensor(0.0001) 194 Training Loss: tensor(0.0005) 195 Training Loss: tensor(0.0002) 196 Training Loss: tensor(0.0006) 197 Training Loss: tensor(0.0003) 198 Training Loss: tensor(0.0005) 199 Training Loss: tensor(0.0005) 200 Training Loss: tensor(0.0004) 201 Training Loss: tensor(0.0004) 202 Training Loss: tensor(0.0002) 203 Training Loss: tensor(0.0006) 204 Training Loss: tensor(0.0003) 205 Training Loss: tensor(0.0006) 206 Training Loss: tensor(0.0002) 207 Training Loss: tensor(0.0003) 208 Training Loss: tensor(0.0001) 209 Training Loss: tensor(0.0002) 210 Training Loss: tensor(0.0003) 211 Training Loss: tensor(0.0002) 212 Training Loss: tensor(0.0002) 213 Training Loss: tensor(0.0003) 214 Training Loss: tensor(0.0004) 215 Training Loss: tensor(0.0002) 216 Training Loss: tensor(0.0003) 217 Training Loss: tensor(0.0002) 218 Training Loss: tensor(0.0004) 219 Training Loss: tensor(0.0002) 220 Training Loss: tensor(0.0002) 221 Training Loss: tensor(0.0002) 222 Training Loss: tensor(0.0002) 223 Training Loss: tensor(0.0002) 224 Training Loss: tensor(0.0002) 225 Training Loss: tensor(0.0002) 226 Training Loss: tensor(0.0002) 227 Training Loss: tensor(0.0002) 228 Training Loss: tensor(0.0002) 229 Training Loss: tensor(0.0002) 230 Training Loss: tensor(0.0005) 231 Training Loss: tensor(0.0008) 232 Training Loss: tensor(0.0014) 233 Training Loss: tensor(0.0014) 234 Training Loss: tensor(0.0012) 235 Training Loss: tensor(0.0006) 236 Training Loss: tensor(0.0003) 237 Training Loss: tensor(0.0006) 238 Training Loss: tensor(0.0011) 239 Training Loss: tensor(0.0025) 240 Training Loss: tensor(0.0033) 241 Training Loss: tensor(0.0030) 242 Training Loss: tensor(0.0015) 243 Training Loss: tensor(0.0004) 244 Training Loss: tensor(0.0012) 245 Training Loss: tensor(0.0025) 246 Training Loss: tensor(0.0035) 247 Training Loss: tensor(0.0024) 248 Training Loss: tensor(0.0013) 249 Training Loss: tensor(0.0005) 250 Training Loss: tensor(0.0018) 251 Training Loss: tensor(0.0023) 252 Training Loss: tensor(0.0031) 253 Training Loss: tensor(0.0019) 254 Training Loss: tensor(0.0015) 255 Training Loss: tensor(0.0007) 256 Training Loss: tensor(0.0022) 257 Training Loss: tensor(0.0016) 258 Training Loss: tensor(0.0014) 259 Training Loss: tensor(0.0010) 260 Training Loss: tensor(0.0031) 261 Training Loss: tensor(0.0030) 262 Training Loss: tensor(0.0013) 263 Training Loss: tensor(0.0012) 264 Training Loss: tensor(0.0030) 265 Training Loss: tensor(0.0039) 266 Training Loss: tensor(0.0031) 267 Training Loss: tensor(0.0010) 268 Training Loss: tensor(0.0020) 269 Training Loss: tensor(0.0019) 270 Training Loss: tensor(0.0031) 271 Training Loss: tensor(0.0007) 272 Training Loss: tensor(0.0033) 273 Training Loss: tensor(0.0018) 274 Training Loss: tensor(0.0019) 275 Training Loss: tensor(0.0005) 276 Training Loss: tensor(0.0020) 277 Training Loss: tensor(0.0010) 278 Training Loss: tensor(0.0012) 279 Training Loss: tensor(0.0005) 280 Training Loss: tensor(0.0013) 281 Training 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In [55]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.6, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.7]$¶
- training with iteration 20000
In [56]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.7, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0002) 1 Training Loss: tensor(0.0002) 2 Training Loss: tensor(0.0002) 3 Training Loss: tensor(5.2093e-05) 4 Training Loss: tensor(9.6378e-05) 5 Training Loss: tensor(0.0002) 6 Training Loss: tensor(0.0002) 7 Training Loss: tensor(0.0002) 8 Training Loss: tensor(0.0002) 9 Training Loss: tensor(0.0002) 10 Training Loss: tensor(0.0002) 11 Training Loss: tensor(8.3679e-05) 12 Training Loss: tensor(0.0002) 13 Training Loss: tensor(0.0001) 14 Training Loss: tensor(7.4251e-05) 15 Training Loss: tensor(0.0002) 16 Training Loss: tensor(9.8402e-05) 17 Training Loss: tensor(0.0001) 18 Training Loss: tensor(0.0001) 19 Training Loss: tensor(0.0002) 20 Training Loss: tensor(0.0001) 21 Training Loss: tensor(0.0001) 22 Training Loss: tensor(0.0002) 23 Training Loss: tensor(0.0001) 24 Training Loss: tensor(0.0002) 25 Training Loss: tensor(0.0002) 26 Training Loss: tensor(0.0003) 27 Training Loss: tensor(0.0001) 28 Training Loss: tensor(0.0002) 29 Training Loss: tensor(0.0002) 30 Training Loss: tensor(0.0002) 31 Training Loss: tensor(9.3685e-05) 32 Training Loss: tensor(0.0001) 33 Training Loss: tensor(0.0002) 34 Training Loss: tensor(0.0001) 35 Training Loss: tensor(8.8408e-05) 36 Training Loss: tensor(0.0001) 37 Training Loss: tensor(0.0002) 38 Training Loss: tensor(0.0001) 39 Training Loss: tensor(9.2693e-05) 40 Training Loss: tensor(0.0002) 41 Training Loss: tensor(0.0002) 42 Training Loss: tensor(0.0001) 43 Training Loss: tensor(0.0001) 44 Training Loss: tensor(0.0002) 45 Training Loss: tensor(0.0002) 46 Training Loss: tensor(0.0002) 47 Training Loss: tensor(0.0001) 48 Training Loss: tensor(0.0001) 49 Training Loss: tensor(0.0002) 50 Training Loss: tensor(0.0001) 51 Training Loss: tensor(0.0002) 52 Training Loss: tensor(0.0001) 53 Training Loss: tensor(0.0004) 54 Training Loss: tensor(0.0003) 55 Training Loss: tensor(0.0002) 56 Training Loss: tensor(0.0001) 57 Training Loss: tensor(0.0002) 58 Training Loss: tensor(0.0002) 59 Training Loss: tensor(5.4366e-05) 60 Training Loss: tensor(0.0001) 61 Training Loss: tensor(0.0002) 62 Training Loss: tensor(0.0002) 63 Training Loss: tensor(0.0001) 64 Training Loss: tensor(0.0001) 65 Training Loss: tensor(0.0002) 66 Training Loss: tensor(0.0001) 67 Training Loss: tensor(0.0001) 68 Training Loss: tensor(9.9417e-05) 69 Training Loss: tensor(0.0002) 70 Training Loss: tensor(0.0001) 71 Training Loss: tensor(0.0001) 72 Training Loss: tensor(7.0768e-05) 73 Training Loss: tensor(0.0001) 74 Training Loss: tensor(0.0001) 75 Training Loss: tensor(6.9397e-05) 76 Training Loss: tensor(9.2689e-05) 77 Training Loss: tensor(0.0002) 78 Training Loss: tensor(0.0001) 79 Training Loss: tensor(7.1493e-05) 80 Training Loss: tensor(0.0002) 81 Training Loss: tensor(0.0002) 82 Training Loss: tensor(0.0002) 83 Training Loss: tensor(8.2056e-05) 84 Training Loss: tensor(0.0001) 85 Training Loss: tensor(0.0002) 86 Training Loss: tensor(0.0002) 87 Training Loss: tensor(0.0001) 88 Training Loss: tensor(0.0002) 89 Training Loss: tensor(0.0003) 90 Training Loss: tensor(0.0005) 91 Training Loss: tensor(0.0005) 92 Training Loss: tensor(0.0002) 93 Training Loss: tensor(0.0002) 94 Training Loss: tensor(0.0002) 95 Training Loss: tensor(0.0003) 96 Training Loss: tensor(0.0001) 97 Training Loss: tensor(0.0001) 98 Training Loss: tensor(0.0003) 99 Training Loss: tensor(0.0003) 100 Training Loss: tensor(0.0005) 101 Training Loss: tensor(0.0002) 102 Training Loss: tensor(0.0002) 103 Training Loss: tensor(0.0003) 104 Training Loss: tensor(0.0004) 105 Training Loss: tensor(0.0004) 106 Training Loss: tensor(0.0002) 107 Training Loss: tensor(0.0002) 108 Training Loss: tensor(0.0005) 109 Training Loss: tensor(0.0004) 110 Training Loss: tensor(0.0002) 111 Training Loss: tensor(0.0002) 112 Training Loss: tensor(0.0004) 113 Training Loss: tensor(0.0005) 114 Training Loss: tensor(0.0002) 115 Training Loss: tensor(6.8795e-05) 116 Training Loss: tensor(0.0003) 117 Training Loss: tensor(0.0003) 118 Training Loss: tensor(0.0003) 119 Training Loss: tensor(0.0002) 120 Training Loss: tensor(0.0002) 121 Training Loss: tensor(0.0004) 122 Training Loss: tensor(0.0006) 123 Training Loss: tensor(0.0003) 124 Training Loss: tensor(0.0003) 125 Training Loss: tensor(0.0003) 126 Training Loss: tensor(0.0003) 127 Training Loss: tensor(0.0003) 128 Training Loss: tensor(0.0001) 129 Training Loss: tensor(0.0002) 130 Training Loss: tensor(0.0001) 131 Training Loss: tensor(0.0002) 132 Training Loss: tensor(0.0001) 133 Training Loss: tensor(0.0002) 134 Training Loss: tensor(0.0003) 135 Training Loss: tensor(0.0003) 136 Training Loss: tensor(0.0001) 137 Training Loss: tensor(0.0001) 138 Training Loss: tensor(0.0003) 139 Training Loss: tensor(0.0003) 140 Training Loss: tensor(7.7589e-05) 141 Training Loss: tensor(0.0002) 142 Training Loss: tensor(0.0004) 143 Training Loss: tensor(0.0004) 144 Training Loss: tensor(0.0002) 145 Training Loss: tensor(7.6267e-05) 146 Training Loss: tensor(0.0001) 147 Training Loss: tensor(0.0002) 148 Training Loss: tensor(0.0003) 149 Training Loss: tensor(0.0001) 150 Training Loss: tensor(0.0001) 151 Training Loss: tensor(0.0001) 152 Training Loss: tensor(0.0003) 153 Training Loss: tensor(0.0002) 154 Training Loss: tensor(7.3773e-05) 155 Training Loss: tensor(0.0001) 156 Training Loss: tensor(0.0002) 157 Training Loss: tensor(0.0001) 158 Training Loss: tensor(4.7412e-05) 159 Training Loss: tensor(0.0001) 160 Training Loss: tensor(0.0002) 161 Training Loss: tensor(0.0001) 162 Training Loss: tensor(6.6297e-05) 163 Training Loss: tensor(0.0001) 164 Training Loss: tensor(0.0002) 165 Training Loss: tensor(0.0001) 166 Training Loss: tensor(8.7315e-05) 167 Training Loss: tensor(8.6007e-05) 168 Training Loss: tensor(0.0002) 169 Training Loss: tensor(0.0001) 170 Training Loss: tensor(0.0002) 171 Training Loss: tensor(6.2750e-05) 172 Training Loss: tensor(0.0001) 173 Training Loss: tensor(0.0002) 174 Training Loss: tensor(7.6913e-05) 175 Training Loss: tensor(0.0001) 176 Training Loss: tensor(0.0001) 177 Training Loss: tensor(0.0002) 178 Training Loss: tensor(0.0002) 179 Training Loss: tensor(9.7741e-05) 180 Training Loss: tensor(0.0001) 181 Training Loss: tensor(0.0001) 182 Training Loss: tensor(0.0002) 183 Training Loss: tensor(9.3293e-05) 184 Training Loss: tensor(9.7829e-05) 185 Training Loss: tensor(9.5793e-05) 186 Training Loss: tensor(0.0001) 187 Training Loss: tensor(7.8796e-05) 188 Training Loss: tensor(8.4150e-05) 189 Training Loss: tensor(0.0001) 190 Training Loss: tensor(0.0002) 191 Training Loss: tensor(5.5958e-05) 192 Training Loss: tensor(9.4485e-05) 193 Training Loss: tensor(0.0001) 194 Training Loss: tensor(0.0001) 195 Training Loss: tensor(0.0001) 196 Training Loss: tensor(0.0001) 197 Training Loss: tensor(0.0002) 198 Training Loss: tensor(9.6305e-05) 199 Training Loss: tensor(8.3108e-05) 200 Training Loss: tensor(9.9459e-05) 201 Training Loss: tensor(0.0001) 202 Training Loss: tensor(8.3625e-05) 203 Training Loss: tensor(0.0001) 204 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Training Loss: tensor(7.7238e-05) 260 Training Loss: tensor(4.6610e-05) 261 Training Loss: tensor(0.0001) 262 Training Loss: tensor(0.0001) 263 Training Loss: tensor(0.0002) 264 Training Loss: tensor(0.0002) 265 Training Loss: tensor(0.0001) 266 Training Loss: tensor(0.0001) 267 Training Loss: tensor(7.3995e-05) 268 Training Loss: tensor(0.0001) 269 Training Loss: tensor(0.0001) 270 Training Loss: tensor(8.0733e-05) 271 Training Loss: tensor(9.3625e-05) 272 Training Loss: tensor(0.0001) 273 Training Loss: tensor(7.4132e-05) 274 Training Loss: tensor(5.1337e-05) 275 Training Loss: tensor(7.0469e-05) 276 Training Loss: tensor(5.1921e-05) 277 Training Loss: tensor(7.5253e-05) 278 Training Loss: tensor(5.3533e-05) 279 Training Loss: tensor(5.2436e-05) 280 Training Loss: tensor(5.6849e-05) 281 Training Loss: tensor(6.1365e-05) 282 Training Loss: tensor(5.4772e-05) 283 Training Loss: tensor(7.9124e-05) 284 Training Loss: tensor(6.7517e-05) 285 Training Loss: tensor(0.0001) 286 Training Loss: 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Training Loss: tensor(0.0002) 19689 Training Loss: tensor(0.0002) 19690 Training Loss: tensor(0.0002) 19691 Training Loss: tensor(0.0004) 19692 Training Loss: tensor(0.0005) 19693 Training Loss: tensor(0.0007) 19694 Training Loss: tensor(0.0006) 19695 Training Loss: tensor(0.0003) 19696 Training Loss: tensor(0.0002) 19697 Training Loss: tensor(0.0005) 19698 Training Loss: tensor(0.0003) 19699 Training Loss: tensor(0.0003) 19700 Training Loss: tensor(9.1691e-05) 19701 Training Loss: tensor(0.0002) 19702 Training Loss: tensor(0.0001) 19703 Training Loss: tensor(0.0001) 19704 Training Loss: tensor(0.0001) 19705 Training Loss: tensor(0.0001) 19706 Training Loss: tensor(9.4631e-05) 19707 Training Loss: tensor(0.0001) 19708 Training Loss: tensor(0.0001) 19709 Training Loss: tensor(7.9173e-05) 19710 Training Loss: tensor(7.8520e-05) 19711 Training Loss: tensor(8.4488e-05) 19712 Training Loss: tensor(0.0001) 19713 Training Loss: tensor(0.0001) 19714 Training Loss: tensor(8.2777e-05) 19715 Training Loss: tensor(5.4833e-05) 19716 Training Loss: tensor(8.6225e-05) 19717 Training Loss: tensor(0.0001) 19718 Training Loss: tensor(6.2242e-05) 19719 Training Loss: tensor(0.0001) 19720 Training Loss: tensor(0.0001) 19721 Training Loss: tensor(0.0002) 19722 Training Loss: tensor(0.0001) 19723 Training Loss: tensor(0.0001) 19724 Training Loss: tensor(0.0001) 19725 Training Loss: tensor(0.0002) 19726 Training Loss: tensor(0.0002) 19727 Training Loss: tensor(0.0002) 19728 Training Loss: tensor(5.6441e-05) 19729 Training Loss: tensor(0.0001) 19730 Training Loss: tensor(0.0001) 19731 Training Loss: tensor(0.0002) 19732 Training Loss: tensor(0.0002) 19733 Training Loss: tensor(7.7924e-05) 19734 Training Loss: tensor(0.0002) 19735 Training Loss: tensor(0.0002) 19736 Training Loss: tensor(0.0002) 19737 Training Loss: tensor(0.0002) 19738 Training Loss: tensor(0.0001) 19739 Training Loss: tensor(0.0002) 19740 Training Loss: tensor(0.0002) 19741 Training Loss: tensor(0.0003) 19742 Training Loss: tensor(0.0002) 19743 Training Loss: tensor(0.0003) 19744 Training Loss: tensor(0.0001) 19745 Training Loss: tensor(0.0002) 19746 Training Loss: tensor(8.5739e-05) 19747 Training Loss: tensor(0.0001) 19748 Training Loss: tensor(0.0001) 19749 Training Loss: tensor(0.0002) 19750 Training Loss: tensor(0.0003) 19751 Training Loss: tensor(0.0002) 19752 Training Loss: tensor(0.0001) 19753 Training Loss: tensor(5.3982e-05) 19754 Training Loss: tensor(0.0001) 19755 Training Loss: tensor(0.0001) 19756 Training Loss: tensor(8.0819e-05) 19757 Training Loss: tensor(9.8595e-05) 19758 Training Loss: tensor(6.3181e-05) 19759 Training Loss: tensor(7.6191e-05) 19760 Training Loss: tensor(0.0001) 19761 Training Loss: tensor(8.5552e-05) 19762 Training Loss: tensor(3.7716e-05) 19763 Training Loss: tensor(6.4023e-05) 19764 Training Loss: tensor(6.9831e-05) 19765 Training Loss: tensor(3.8711e-05) 19766 Training Loss: tensor(6.9023e-05) 19767 Training Loss: tensor(7.4776e-05) 19768 Training Loss: tensor(0.0001) 19769 Training Loss: tensor(0.0001) 19770 Training Loss: tensor(7.5355e-05) 19771 Training Loss: tensor(4.9898e-05) 19772 Training Loss: tensor(7.4059e-05) 19773 Training Loss: tensor(8.3351e-05) 19774 Training Loss: tensor(9.6168e-05) 19775 Training Loss: tensor(7.4151e-05) 19776 Training Loss: tensor(5.5586e-05) 19777 Training Loss: tensor(5.4680e-05) 19778 Training Loss: tensor(3.8904e-05) 19779 Training Loss: tensor(4.4566e-05) 19780 Training Loss: tensor(5.3904e-05) 19781 Training Loss: tensor(3.2274e-05) 19782 Training Loss: tensor(4.0093e-05) 19783 Training Loss: tensor(4.0383e-05) 19784 Training Loss: tensor(3.3276e-05) 19785 Training Loss: tensor(5.1623e-05) 19786 Training Loss: tensor(0.0001) 19787 Training Loss: tensor(9.1354e-05) 19788 Training Loss: tensor(7.0114e-05) 19789 Training Loss: tensor(7.3189e-05) 19790 Training Loss: tensor(6.2893e-05) 19791 Training Loss: tensor(7.1995e-05) 19792 Training Loss: tensor(4.5423e-05) 19793 Training Loss: tensor(0.0001) 19794 Training Loss: tensor(6.0094e-05) 19795 Training Loss: tensor(0.0001) 19796 Training Loss: tensor(0.0001) 19797 Training Loss: tensor(9.0386e-05) 19798 Training Loss: tensor(7.2469e-05) 19799 Training Loss: tensor(5.8478e-05) 19800 Training Loss: tensor(8.0385e-05) 19801 Training Loss: tensor(8.0271e-05) 19802 Training Loss: tensor(0.0001) 19803 Training Loss: tensor(0.0001) 19804 Training Loss: tensor(0.0001) 19805 Training Loss: tensor(7.3305e-05) 19806 Training Loss: tensor(0.0001) 19807 Training Loss: tensor(0.0002) 19808 Training Loss: tensor(0.0002) 19809 Training Loss: tensor(0.0001) 19810 Training Loss: tensor(5.7022e-05) 19811 Training Loss: tensor(9.8184e-05) 19812 Training Loss: tensor(0.0002) 19813 Training Loss: tensor(0.0004) 19814 Training Loss: tensor(0.0005) 19815 Training Loss: tensor(0.0006) 19816 Training Loss: tensor(0.0004) 19817 Training Loss: tensor(0.0001) 19818 Training Loss: tensor(0.0002) 19819 Training Loss: tensor(0.0005) 19820 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Training Loss: tensor(0.0003) 19903 Training Loss: tensor(0.0003) 19904 Training Loss: tensor(0.0003) 19905 Training Loss: tensor(0.0001) 19906 Training Loss: tensor(0.0003) 19907 Training Loss: tensor(0.0002) 19908 Training Loss: tensor(0.0004) 19909 Training Loss: tensor(0.0002) 19910 Training Loss: tensor(0.0003) 19911 Training Loss: tensor(0.0003) 19912 Training Loss: tensor(0.0005) 19913 Training Loss: tensor(0.0003) 19914 Training Loss: tensor(0.0004) 19915 Training Loss: tensor(0.0004) 19916 Training Loss: tensor(0.0004) 19917 Training Loss: tensor(0.0003) 19918 Training Loss: tensor(0.0005) 19919 Training Loss: tensor(0.0004) 19920 Training Loss: tensor(0.0004) 19921 Training Loss: tensor(0.0008) 19922 Training Loss: tensor(0.0010) 19923 Training Loss: tensor(0.0011) 19924 Training Loss: tensor(0.0004) 19925 Training Loss: tensor(0.0003) 19926 Training Loss: tensor(0.0005) 19927 Training Loss: tensor(0.0006) 19928 Training Loss: tensor(0.0003) 19929 Training Loss: 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Training Loss: tensor(0.0004) 19958 Training Loss: tensor(0.0006) 19959 Training Loss: tensor(0.0004) 19960 Training Loss: tensor(0.0004) 19961 Training Loss: tensor(0.0006) 19962 Training Loss: tensor(0.0007) 19963 Training Loss: tensor(0.0012) 19964 Training Loss: tensor(0.0008) 19965 Training Loss: tensor(0.0008) 19966 Training Loss: tensor(0.0005) 19967 Training Loss: tensor(0.0014) 19968 Training Loss: tensor(0.0007) 19969 Training Loss: tensor(0.0010) 19970 Training Loss: tensor(0.0004) 19971 Training Loss: tensor(0.0007) 19972 Training Loss: tensor(0.0006) 19973 Training Loss: tensor(0.0007) 19974 Training Loss: tensor(0.0007) 19975 Training Loss: tensor(0.0008) 19976 Training Loss: tensor(0.0011) 19977 Training Loss: tensor(0.0006) 19978 Training Loss: tensor(0.0005) 19979 Training Loss: tensor(0.0005) 19980 Training Loss: tensor(0.0005) 19981 Training Loss: tensor(0.0005) 19982 Training Loss: tensor(0.0006) 19983 Training Loss: tensor(0.0003) 19984 Training Loss: tensor(0.0007) 19985 Training Loss: tensor(0.0004) 19986 Training Loss: tensor(0.0005) 19987 Training Loss: tensor(0.0004) 19988 Training Loss: tensor(0.0012) 19989 Training Loss: tensor(0.0013) 19990 Training Loss: tensor(0.0015) 19991 Training Loss: tensor(0.0008) 19992 Training Loss: tensor(0.0010) 19993 Training Loss: tensor(0.0005) 19994 Training Loss: tensor(0.0007) 19995 Training Loss: tensor(0.0010) 19996 Training Loss: tensor(0.0007) 19997 Training Loss: tensor(0.0015) 19998 Training Loss: tensor(0.0006) 19999 Training Loss: tensor(0.0011)
In [57]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.7, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.8]$¶
- training with iteration 20000
In [58]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.8, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0005) 1 Training Loss: tensor(0.0015) 2 Training Loss: tensor(0.0006) 3 Training Loss: tensor(0.0012) 4 Training Loss: tensor(0.0007) 5 Training Loss: tensor(0.0005) 6 Training Loss: tensor(0.0006) 7 Training Loss: tensor(0.0008) 8 Training Loss: tensor(0.0007) 9 Training Loss: tensor(0.0006) 10 Training Loss: tensor(0.0009) 11 Training Loss: tensor(0.0002) 12 Training Loss: tensor(0.0009) 13 Training Loss: tensor(0.0002) 14 Training Loss: tensor(0.0004) 15 Training Loss: tensor(0.0004) 16 Training Loss: tensor(0.0003) 17 Training Loss: tensor(0.0005) 18 Training Loss: tensor(0.0003) 19 Training Loss: tensor(0.0003) 20 Training Loss: tensor(0.0003) 21 Training Loss: tensor(0.0005) 22 Training Loss: tensor(0.0004) 23 Training Loss: tensor(0.0004) 24 Training Loss: tensor(0.0004) 25 Training Loss: tensor(0.0003) 26 Training Loss: tensor(0.0004) 27 Training Loss: tensor(0.0003) 28 Training Loss: tensor(0.0005) 29 Training Loss: tensor(0.0002) 30 Training Loss: tensor(0.0004) 31 Training Loss: tensor(0.0004) 32 Training Loss: tensor(0.0004) 33 Training Loss: tensor(0.0005) 34 Training Loss: tensor(0.0004) 35 Training Loss: tensor(0.0004) 36 Training Loss: tensor(0.0005) 37 Training Loss: tensor(0.0005) 38 Training Loss: tensor(0.0005) 39 Training Loss: tensor(0.0003) 40 Training Loss: tensor(0.0005) 41 Training Loss: tensor(0.0004) 42 Training Loss: tensor(0.0004) 43 Training Loss: tensor(0.0006) 44 Training Loss: tensor(0.0003) 45 Training Loss: tensor(0.0005) 46 Training Loss: tensor(0.0004) 47 Training Loss: tensor(0.0008) 48 Training Loss: tensor(0.0008) 49 Training Loss: tensor(0.0003) 50 Training Loss: tensor(0.0004) 51 Training Loss: tensor(0.0006) 52 Training Loss: tensor(0.0004) 53 Training Loss: tensor(0.0004) 54 Training Loss: tensor(0.0004) 55 Training Loss: tensor(0.0007) 56 Training Loss: tensor(0.0007) 57 Training Loss: tensor(0.0003) 58 Training Loss: tensor(0.0002) 59 Training Loss: tensor(0.0004) 60 Training Loss: tensor(0.0006) 61 Training Loss: tensor(0.0003) 62 Training Loss: tensor(0.0004) 63 Training Loss: tensor(0.0002) 64 Training Loss: tensor(0.0002) 65 Training Loss: tensor(0.0003) 66 Training Loss: tensor(0.0002) 67 Training Loss: tensor(0.0003) 68 Training Loss: tensor(0.0002) 69 Training Loss: tensor(0.0003) 70 Training Loss: tensor(0.0002) 71 Training Loss: tensor(0.0003) 72 Training Loss: tensor(0.0002) 73 Training Loss: tensor(0.0002) 74 Training Loss: tensor(0.0002) 75 Training Loss: tensor(0.0003) 76 Training Loss: tensor(0.0002) 77 Training Loss: tensor(0.0002) 78 Training Loss: tensor(0.0002) 79 Training Loss: tensor(0.0002) 80 Training Loss: tensor(0.0004) 81 Training Loss: tensor(0.0003) 82 Training Loss: tensor(0.0005) 83 Training Loss: tensor(0.0002) 84 Training Loss: tensor(0.0004) 85 Training Loss: tensor(0.0005) 86 Training Loss: tensor(0.0007) 87 Training Loss: tensor(0.0005) 88 Training Loss: tensor(0.0003) 89 Training Loss: tensor(0.0011) 90 Training Loss: tensor(0.0018) 91 Training Loss: tensor(0.0018) 92 Training Loss: tensor(0.0009) 93 Training Loss: tensor(0.0014) 94 Training Loss: tensor(0.0028) 95 Training Loss: tensor(0.0026) 96 Training Loss: tensor(0.0010) 97 Training Loss: tensor(0.0016) 98 Training Loss: tensor(0.0034) 99 Training Loss: tensor(0.0013) 100 Training Loss: tensor(0.0016) 101 Training Loss: tensor(0.0028) 102 Training Loss: tensor(0.0019) 103 Training Loss: tensor(0.0009) 104 Training Loss: tensor(0.0009) 105 Training Loss: tensor(0.0016) 106 Training Loss: tensor(0.0011) 107 Training Loss: tensor(0.0014) 108 Training Loss: tensor(0.0028) 109 Training Loss: tensor(0.0019) 110 Training Loss: tensor(0.0008) 111 Training Loss: tensor(0.0015) 112 Training Loss: tensor(0.0008) 113 Training Loss: tensor(0.0017) 114 Training Loss: tensor(0.0026) 115 Training Loss: tensor(0.0015) 116 Training Loss: tensor(0.0006) 117 Training Loss: tensor(0.0020) 118 Training Loss: tensor(0.0009) 119 Training Loss: tensor(0.0006) 120 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In [59]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.8, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 0.9]$¶
- training with iteration 20000
In [60]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=0.9, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(0.0001) 1 Training Loss: tensor(9.1912e-05) 2 Training Loss: tensor(0.0002) 3 Training Loss: tensor(0.0001) 4 Training Loss: tensor(0.0001) 5 Training Loss: tensor(0.0001) 6 Training Loss: tensor(0.0001) 7 Training Loss: tensor(0.0002) 8 Training Loss: tensor(0.0002) 9 Training Loss: tensor(6.4956e-05) 10 Training Loss: tensor(0.0001) 11 Training Loss: tensor(0.0002) 12 Training Loss: tensor(9.4965e-05) 13 Training Loss: tensor(7.1151e-05) 14 Training Loss: tensor(0.0002) 15 Training Loss: tensor(0.0002) 16 Training Loss: tensor(5.1615e-05) 17 Training Loss: tensor(9.8498e-05) 18 Training Loss: tensor(0.0001) 19 Training Loss: tensor(0.0001) 20 Training Loss: tensor(4.2038e-05) 21 Training Loss: tensor(0.0001) 22 Training Loss: tensor(0.0002) 23 Training Loss: tensor(8.0944e-05) 24 Training Loss: tensor(6.8331e-05) 25 Training Loss: tensor(7.8201e-05) 26 Training Loss: tensor(8.0134e-05) 27 Training Loss: tensor(6.1361e-05) 28 Training Loss: tensor(6.5892e-05) 29 Training Loss: tensor(4.9047e-05) 30 Training Loss: tensor(3.8860e-05) 31 Training Loss: tensor(4.0136e-05) 32 Training Loss: tensor(6.1833e-05) 33 Training Loss: tensor(5.0801e-05) 34 Training Loss: tensor(4.9548e-05) 35 Training Loss: tensor(4.1024e-05) 36 Training Loss: tensor(8.4328e-05) 37 Training Loss: tensor(4.8057e-05) 38 Training Loss: tensor(0.0001) 39 Training Loss: tensor(7.1755e-05) 40 Training Loss: tensor(9.1068e-05) 41 Training Loss: tensor(0.0002) 42 Training Loss: tensor(0.0001) 43 Training Loss: tensor(0.0001) 44 Training Loss: tensor(7.1323e-05) 45 Training Loss: tensor(9.6788e-05) 46 Training Loss: tensor(8.1603e-05) 47 Training Loss: tensor(7.9030e-05) 48 Training Loss: tensor(5.2848e-05) 49 Training Loss: tensor(6.0522e-05) 50 Training Loss: tensor(6.1167e-05) 51 Training Loss: tensor(4.5411e-05) 52 Training Loss: tensor(9.9939e-05) 53 Training Loss: tensor(3.6768e-05) 54 Training Loss: tensor(5.9047e-05) 55 Training Loss: tensor(6.2690e-05) 56 Training Loss: tensor(5.6268e-05) 57 Training Loss: tensor(4.4208e-05) 58 Training Loss: tensor(4.4149e-05) 59 Training Loss: tensor(4.9967e-05) 60 Training Loss: tensor(7.8192e-05) 61 Training Loss: tensor(5.3355e-05) 62 Training Loss: tensor(5.4047e-05) 63 Training Loss: tensor(8.3828e-05) 64 Training Loss: tensor(0.0001) 65 Training Loss: tensor(7.1506e-05) 66 Training Loss: tensor(3.5494e-05) 67 Training Loss: tensor(8.0744e-05) 68 Training Loss: tensor(0.0001) 69 Training Loss: tensor(0.0001) 70 Training Loss: tensor(4.5549e-05) 71 Training Loss: tensor(4.9658e-05) 72 Training Loss: tensor(8.1069e-05) 73 Training Loss: tensor(4.6404e-05) 74 Training Loss: tensor(3.0987e-05) 75 Training Loss: tensor(4.5088e-05) 76 Training Loss: tensor(3.9666e-05) 77 Training Loss: tensor(5.1369e-05) 78 Training Loss: tensor(3.0277e-05) 79 Training Loss: tensor(4.9353e-05) 80 Training Loss: tensor(2.2014e-05) 81 Training Loss: tensor(5.4945e-05) 82 Training Loss: tensor(4.8712e-05) 83 Training Loss: tensor(3.5047e-05) 84 Training Loss: tensor(8.6728e-05) 85 Training Loss: tensor(2.2802e-05) 86 Training Loss: tensor(6.7267e-05) 87 Training Loss: tensor(4.2744e-05) 88 Training Loss: tensor(5.6953e-05) 89 Training Loss: tensor(2.0174e-05) 90 Training Loss: tensor(2.3383e-05) 91 Training Loss: tensor(2.8615e-05) 92 Training Loss: tensor(0.0001) 93 Training Loss: tensor(8.4354e-05) 94 Training Loss: tensor(4.3390e-05) 95 Training Loss: tensor(7.4285e-05) 96 Training Loss: tensor(9.6268e-05) 97 Training Loss: tensor(9.9293e-05) 98 Training Loss: tensor(6.5211e-05) 99 Training Loss: tensor(8.0190e-05) 100 Training Loss: tensor(7.3818e-05) 101 Training Loss: tensor(0.0001) 102 Training Loss: tensor(0.0001) 103 Training Loss: tensor(8.4571e-05) 104 Training Loss: tensor(0.0002) 105 Training Loss: tensor(9.2035e-05) 106 Training Loss: tensor(0.0002) 107 Training Loss: tensor(0.0002) 108 Training Loss: tensor(0.0001) 109 Training Loss: tensor(9.8040e-05) 110 Training Loss: tensor(0.0002) 111 Training Loss: tensor(0.0002) 112 Training Loss: tensor(0.0004) 113 Training Loss: tensor(7.3936e-05) 114 Training Loss: tensor(0.0003) 115 Training Loss: tensor(0.0004) 116 Training Loss: tensor(0.0002) 117 Training Loss: tensor(0.0002) 118 Training Loss: tensor(0.0004) 119 Training Loss: tensor(0.0007) 120 Training Loss: tensor(0.0006) 121 Training Loss: tensor(0.0002) 122 Training Loss: tensor(5.3965e-05) 123 Training Loss: tensor(0.0003) 124 Training Loss: tensor(0.0006) 125 Training Loss: tensor(0.0012) 126 Training Loss: tensor(0.0010) 127 Training Loss: tensor(0.0004) 128 Training Loss: tensor(0.0004) 129 Training Loss: tensor(0.0008) 130 Training Loss: tensor(0.0007) 131 Training Loss: tensor(0.0007) 132 Training Loss: tensor(0.0006) 133 Training Loss: tensor(0.0005) 134 Training Loss: tensor(0.0007) 135 Training Loss: tensor(0.0006) 136 Training Loss: tensor(0.0005) 137 Training Loss: tensor(0.0004) 138 Training Loss: tensor(0.0009) 139 Training Loss: tensor(0.0003) 140 Training Loss: tensor(0.0004) 141 Training Loss: tensor(0.0004) 142 Training Loss: tensor(0.0007) 143 Training Loss: tensor(0.0005) 144 Training Loss: tensor(0.0004) 145 Training Loss: tensor(0.0003) 146 Training Loss: tensor(0.0007) 147 Training Loss: tensor(0.0015) 148 Training Loss: tensor(0.0009) 149 Training Loss: tensor(0.0006) 150 Training Loss: tensor(0.0007) 151 Training Loss: tensor(0.0013) 152 Training Loss: tensor(0.0004) 153 Training Loss: tensor(0.0008) 154 Training Loss: tensor(0.0015) 155 Training Loss: tensor(0.0007) 156 Training Loss: tensor(0.0006) 157 Training Loss: tensor(0.0015) 158 Training Loss: tensor(0.0020) 159 Training Loss: tensor(0.0006) 160 Training Loss: tensor(0.0003) 161 Training Loss: tensor(0.0009) 162 Training Loss: tensor(0.0004) 163 Training Loss: tensor(0.0004) 164 Training Loss: tensor(0.0004) 165 Training Loss: tensor(0.0004) 166 Training Loss: tensor(0.0004) 167 Training Loss: tensor(0.0003) 168 Training Loss: tensor(0.0005) 169 Training Loss: tensor(0.0003) 170 Training Loss: tensor(0.0004) 171 Training Loss: tensor(0.0007) 172 Training Loss: tensor(0.0002) 173 Training Loss: tensor(0.0004) 174 Training Loss: tensor(0.0004) 175 Training Loss: tensor(0.0003) 176 Training Loss: tensor(0.0003) 177 Training Loss: tensor(0.0003) 178 Training Loss: tensor(0.0003) 179 Training Loss: tensor(0.0003) 180 Training Loss: tensor(0.0003) 181 Training Loss: tensor(0.0004) 182 Training Loss: tensor(0.0001) 183 Training Loss: tensor(0.0003) 184 Training Loss: tensor(0.0003) 185 Training Loss: tensor(0.0002) 186 Training Loss: tensor(0.0003) 187 Training Loss: tensor(0.0001) 188 Training Loss: tensor(0.0004) 189 Training Loss: tensor(0.0001) 190 Training Loss: tensor(0.0003) 191 Training Loss: tensor(0.0002) 192 Training Loss: tensor(0.0001) 193 Training Loss: tensor(0.0002) 194 Training Loss: tensor(8.9691e-05) 195 Training Loss: tensor(0.0002) 196 Training Loss: tensor(0.0001) 197 Training Loss: tensor(0.0001) 198 Training Loss: 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227 Training Loss: tensor(0.0002) 228 Training Loss: tensor(7.8841e-05) 229 Training Loss: tensor(0.0001) 230 Training Loss: tensor(0.0002) 231 Training Loss: tensor(0.0002) 232 Training Loss: tensor(0.0003) 233 Training Loss: tensor(0.0002) 234 Training Loss: tensor(0.0002) 235 Training Loss: tensor(0.0002) 236 Training Loss: tensor(0.0003) 237 Training Loss: tensor(0.0001) 238 Training Loss: tensor(0.0008) 239 Training Loss: tensor(0.0012) 240 Training Loss: tensor(0.0016) 241 Training Loss: tensor(0.0008) 242 Training Loss: tensor(0.0005) 243 Training Loss: tensor(0.0007) 244 Training Loss: tensor(0.0008) 245 Training Loss: tensor(0.0004) 246 Training Loss: tensor(0.0005) 247 Training Loss: tensor(0.0004) 248 Training Loss: tensor(0.0003) 249 Training Loss: tensor(0.0004) 250 Training Loss: tensor(0.0004) 251 Training Loss: tensor(0.0002) 252 Training Loss: tensor(0.0002) 253 Training Loss: tensor(0.0003) 254 Training Loss: tensor(0.0002) 255 Training Loss: tensor(0.0002) 256 Training Loss: tensor(0.0002) 257 Training Loss: tensor(0.0001) 258 Training Loss: tensor(0.0001) 259 Training Loss: tensor(0.0003) 260 Training Loss: tensor(0.0003) 261 Training Loss: tensor(0.0002) 262 Training Loss: tensor(0.0001) 263 Training Loss: tensor(0.0001) 264 Training Loss: tensor(0.0001) 265 Training Loss: tensor(7.7675e-05) 266 Training Loss: tensor(0.0001) 267 Training Loss: tensor(0.0001) 268 Training Loss: tensor(9.1000e-05) 269 Training Loss: tensor(0.0001) 270 Training Loss: tensor(6.8223e-05) 271 Training Loss: tensor(0.0001) 272 Training Loss: tensor(7.9521e-05) 273 Training Loss: tensor(9.9802e-05) 274 Training Loss: tensor(6.7615e-05) 275 Training Loss: tensor(8.3524e-05) 276 Training Loss: tensor(7.2144e-05) 277 Training Loss: tensor(0.0001) 278 Training Loss: tensor(5.5362e-05) 279 Training Loss: tensor(5.4495e-05) 280 Training Loss: tensor(9.9999e-05) 281 Training Loss: tensor(4.3672e-05) 282 Training Loss: tensor(6.0468e-05) 283 Training Loss: tensor(0.0001) 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In [61]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 0.9, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.5.2 Fitting model with Time-adaptive approach1 - 1st time step for $t \in [0, 1]$¶
- training with iteration 20000
In [62]:
# Setting the number of iterations
num_epochs = 20000
#prev_val_loss = 9999999.0
# Iterating through epochs
for epoch in range(num_epochs):
# Resetting gradients to zero
optimizer.zero_grad()
# Loss calculation based on boundary conditions
input_x_bc = Variable(x_bc.float(), requires_grad=False).to(device)
#input_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
input_t_bc = Variable(t_bc.float(), requires_grad=False).to(device)
#input_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
target_u_bc = Variable(u_bc.float(), requires_grad=False).to(device)
#target_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
target_u_initial1 = Variable(u_initial1.float(), requires_grad=False).to(device)
target_u_initial2 = Variable(u_initial2.float(), requires_grad=False).to(device)
# Getting network output for boundary condition and initial conditions
output_bc = net(input_x_bc, input_t_bc)
output_initial1 = net(input_t_bc, target_u_initial1)
output_initial2 = net(input_t_bc, target_u_initial2)
mse_u = mse_cost_function(output_bc, target_u_bc)
mse_initial1 = mse_cost_function(u_initial1, target_u_initial1)
mse_initial2 = mse_cost_function(u_initial2, target_u_initial2)
# Loss calculation based on partial differential equation (PDE)
collocation_x = np.random.uniform(low= -2.0, high=2.0, size=(500, 1))
collocation_t = np.random.uniform(low=0.0, high=1.0, size=(500, 1))
all_zeros_target = np.zeros((500, 1))
input_x_collocation = Variable(torch.from_numpy(collocation_x).float(), requires_grad=True).to(device)
input_t_collocation = Variable(torch.from_numpy(collocation_t).float(), requires_grad=True).to(device)
target_all_zeros = Variable(torch.from_numpy(all_zeros_target).float(), requires_grad=False).to(device)
# Getting network output for PDE
output_f = pde_loss(input_x_collocation, input_t_collocation, net)
mse_f = mse_cost_function(output_f, target_all_zeros)
# Combining the loss functions
total_loss = mse_u + mse_f + mse_initial1 + mse_initial2
# Backward propagation for computing gradients
total_loss.backward()
# Optimizer step to update parameters
optimizer.step()
with torch.autograd.no_grad():
print(epoch, "Training Loss:", total_loss.data)
0 Training Loss: tensor(3.8828e-05) 1 Training Loss: tensor(2.7742e-05) 2 Training Loss: tensor(1.3316e-05) 3 Training Loss: tensor(3.4266e-05) 4 Training Loss: tensor(4.4673e-05) 5 Training Loss: tensor(2.6217e-05) 6 Training Loss: tensor(3.0993e-05) 7 Training Loss: tensor(6.5519e-05) 8 Training Loss: tensor(4.3587e-05) 9 Training Loss: tensor(3.5874e-05) 10 Training Loss: tensor(5.1155e-05) 11 Training Loss: tensor(3.7891e-05) 12 Training Loss: tensor(3.9540e-05) 13 Training Loss: tensor(2.8922e-05) 14 Training Loss: tensor(5.7987e-05) 15 Training Loss: tensor(3.6609e-05) 16 Training Loss: tensor(4.2687e-05) 17 Training Loss: tensor(5.4379e-05) 18 Training Loss: tensor(2.6861e-05) 19 Training Loss: tensor(4.1448e-05) 20 Training Loss: tensor(4.6619e-05) 21 Training Loss: tensor(5.1882e-05) 22 Training Loss: tensor(7.5118e-05) 23 Training Loss: tensor(2.1631e-05) 24 Training Loss: tensor(4.7988e-05) 25 Training Loss: tensor(4.7285e-05) 26 Training Loss: tensor(2.6377e-05) 27 Training Loss: tensor(5.4368e-05) 28 Training Loss: tensor(3.9109e-05) 29 Training Loss: tensor(3.9151e-05) 30 Training Loss: tensor(4.8353e-05) 31 Training Loss: tensor(2.9224e-05) 32 Training Loss: tensor(4.6023e-05) 33 Training Loss: tensor(3.6256e-05) 34 Training Loss: tensor(2.7328e-05) 35 Training Loss: tensor(3.9406e-05) 36 Training Loss: tensor(2.7782e-05) 37 Training Loss: tensor(2.9946e-05) 38 Training Loss: tensor(4.8577e-05) 39 Training Loss: tensor(1.8431e-05) 40 Training Loss: tensor(3.5576e-05) 41 Training Loss: tensor(2.5072e-05) 42 Training Loss: tensor(1.8075e-05) 43 Training Loss: tensor(3.2716e-05) 44 Training Loss: tensor(2.3149e-05) 45 Training Loss: tensor(2.5300e-05) 46 Training Loss: tensor(4.7465e-05) 47 Training Loss: tensor(2.4237e-05) 48 Training Loss: tensor(7.3109e-05) 49 Training Loss: tensor(2.2444e-05) 50 Training Loss: tensor(3.5296e-05) 51 Training Loss: tensor(4.6996e-05) 52 Training Loss: tensor(4.2444e-05) 53 Training Loss: tensor(6.4271e-05) 54 Training Loss: tensor(3.4833e-05) 55 Training Loss: tensor(5.1763e-05) 56 Training Loss: tensor(5.1385e-05) 57 Training Loss: tensor(2.2193e-05) 58 Training Loss: tensor(7.4169e-05) 59 Training Loss: tensor(3.9427e-05) 60 Training Loss: tensor(3.1493e-05) 61 Training Loss: tensor(0.0001) 62 Training Loss: tensor(3.1538e-05) 63 Training Loss: tensor(5.3966e-05) 64 Training Loss: tensor(7.2877e-05) 65 Training Loss: tensor(3.6345e-05) 66 Training Loss: tensor(0.0002) 67 Training Loss: tensor(4.9773e-05) 68 Training Loss: tensor(6.8315e-05) 69 Training Loss: tensor(7.7622e-05) 70 Training Loss: tensor(4.6500e-05) 71 Training Loss: tensor(6.6561e-05) 72 Training Loss: tensor(7.5772e-05) 73 Training Loss: tensor(8.2839e-05) 74 Training Loss: tensor(7.5742e-05) 75 Training Loss: tensor(6.2220e-05) 76 Training Loss: tensor(8.6531e-05) 77 Training Loss: tensor(4.9772e-05) 78 Training Loss: tensor(4.1266e-05) 79 Training Loss: tensor(7.0572e-05) 80 Training Loss: tensor(4.5121e-05) 81 Training Loss: tensor(5.4265e-05) 82 Training Loss: tensor(5.4521e-05) 83 Training Loss: tensor(6.7940e-05) 84 Training Loss: tensor(6.2483e-05) 85 Training Loss: tensor(6.6536e-05) 86 Training Loss: tensor(9.2785e-05) 87 Training Loss: tensor(4.3976e-05) 88 Training Loss: tensor(0.0001) 89 Training Loss: tensor(4.8723e-05) 90 Training Loss: tensor(6.4746e-05) 91 Training Loss: tensor(6.1846e-05) 92 Training Loss: tensor(5.2067e-05) 93 Training Loss: tensor(5.0883e-05) 94 Training Loss: tensor(2.1058e-05) 95 Training Loss: tensor(3.8201e-05) 96 Training Loss: tensor(4.8391e-05) 97 Training Loss: tensor(3.0169e-05) 98 Training Loss: tensor(3.2186e-05) 99 Training Loss: tensor(4.5767e-05) 100 Training Loss: tensor(4.3202e-05) 101 Training Loss: tensor(2.9933e-05) 102 Training Loss: tensor(3.5651e-05) 103 Training Loss: tensor(6.3609e-05) 104 Training Loss: tensor(2.2499e-05) 105 Training Loss: tensor(4.3132e-05) 106 Training Loss: tensor(4.6163e-05) 107 Training Loss: tensor(2.6871e-05) 108 Training Loss: tensor(4.5737e-05) 109 Training Loss: tensor(3.9240e-05) 110 Training Loss: tensor(4.8243e-05) 111 Training Loss: tensor(4.2492e-05) 112 Training Loss: tensor(4.9269e-05) 113 Training Loss: tensor(5.8677e-05) 114 Training Loss: tensor(3.1342e-05) 115 Training Loss: tensor(4.0310e-05) 116 Training Loss: tensor(4.9647e-05) 117 Training Loss: tensor(3.3037e-05) 118 Training Loss: tensor(6.1772e-05) 119 Training Loss: tensor(4.4506e-05) 120 Training Loss: tensor(2.1710e-05) 121 Training Loss: tensor(6.0717e-05) 122 Training Loss: tensor(1.9735e-05) 123 Training Loss: tensor(4.1793e-05) 124 Training Loss: tensor(5.9508e-05) 125 Training Loss: tensor(3.6693e-05) 126 Training Loss: tensor(4.0964e-05) 127 Training Loss: tensor(6.9839e-05) 128 Training Loss: tensor(2.7009e-05) 129 Training Loss: tensor(3.7354e-05) 130 Training Loss: tensor(3.8327e-05) 131 Training Loss: tensor(3.7989e-05) 132 Training Loss: tensor(3.0366e-05) 133 Training Loss: tensor(4.6310e-05) 134 Training Loss: tensor(2.8943e-05) 135 Training Loss: tensor(2.3869e-05) 136 Training Loss: tensor(2.6898e-05) 137 Training Loss: tensor(2.2379e-05) 138 Training Loss: tensor(2.3287e-05) 139 Training Loss: tensor(1.9795e-05) 140 Training Loss: tensor(3.0073e-05) 141 Training Loss: tensor(2.2459e-05) 142 Training Loss: tensor(1.3310e-05) 143 Training Loss: tensor(3.2328e-05) 144 Training Loss: tensor(3.0974e-05) 145 Training Loss: tensor(3.3925e-05) 146 Training Loss: tensor(3.1555e-05) 147 Training Loss: tensor(3.1022e-05) 148 Training Loss: tensor(2.7392e-05) 149 Training Loss: tensor(3.0919e-05) 150 Training Loss: tensor(1.3909e-05) 151 Training Loss: tensor(3.0203e-05) 152 Training Loss: tensor(2.9008e-05) 153 Training Loss: tensor(2.1797e-05) 154 Training Loss: tensor(3.7809e-05) 155 Training Loss: tensor(1.1177e-05) 156 Training Loss: tensor(2.5938e-05) 157 Training Loss: tensor(1.7185e-05) 158 Training Loss: tensor(1.3896e-05) 159 Training Loss: tensor(1.9998e-05) 160 Training Loss: 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291 Training Loss: tensor(0.0001) 292 Training Loss: tensor(7.9651e-05) 293 Training Loss: tensor(6.9795e-05) 294 Training Loss: tensor(7.9692e-05) 295 Training Loss: tensor(6.8758e-05) 296 Training Loss: tensor(5.5430e-05) 297 Training Loss: tensor(5.6979e-05) 298 Training Loss: tensor(4.8672e-05) 299 Training Loss: tensor(7.8549e-05) 300 Training Loss: tensor(5.1357e-05) 301 Training Loss: tensor(0.0001) 302 Training Loss: tensor(9.8099e-05) 303 Training Loss: tensor(7.2824e-05) 304 Training Loss: tensor(6.3905e-05) 305 Training Loss: tensor(7.3380e-05) 306 Training Loss: tensor(7.9714e-05) 307 Training Loss: tensor(0.0001) 308 Training Loss: tensor(7.7647e-05) 309 Training Loss: tensor(0.0001) 310 Training Loss: tensor(8.6012e-05) 311 Training Loss: tensor(0.0001) 312 Training Loss: tensor(8.6242e-05) 313 Training Loss: tensor(6.8790e-05) 314 Training Loss: tensor(0.0001) 315 Training Loss: tensor(6.2042e-05) 316 Training Loss: tensor(0.0001) 317 Training Loss: tensor(0.0001) 318 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In [63]:
# Plotting the results
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
import numpy as np
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# Creating meshgrid for visualization
x_visual = np.arange(-2.0 , 2.0, 0.002)
t_visual = np.arange(0, 1.0, 0.002)
ms_x_visual, ms_t_visual = np.meshgrid(x_visual, t_visual)
## np.ravel returns a flattened and contiguous 1-D array containing the elements of the input.
## A copy is created only if necessary.
x_visual = np.ravel(ms_x_visual).reshape(-1, 1)
t_visual = np.ravel(ms_t_visual).reshape(-1, 1)
# Converting to PyTorch tensors
pt_x_visual = Variable(torch.from_numpy(x_visual).float(), requires_grad=True).to(device)
pt_t_visual = Variable(torch.from_numpy(t_visual).float(), requires_grad=True).to(device)
# Obtaining network predictions
## In the function net(pt_x_visualization, pt_t_visualization), the training process updates the parameters represented by theta.
pt_u_visual = net(pt_x_visual, pt_t_visual)
u_visual = pt_u_visual.data.cpu().numpy()
ms_u_visual = u_visual.reshape(ms_x_visual.shape)
# Plotting the surface
surf = ax.plot_surface(ms_x_visual, ms_t_visual, ms_u_visual, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.0003f'))
ax.set_xlabel("x_axis")
ax.set_ylabel("t_axis")
# Adding colorbar for reference
fig.colorbar(surf, shrink=1, aspect=5)
# Displaying the plot
plt.show()
3.6.3 Comparing Training set with Actual Solution Graph¶
In [206]:
import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
# Parameter setting
terminal_time = 1 # terminal time
num_spatial_values = 80 # spatial values for FEM
num_time_values = 1600 # time values/hops; (time_values ~ spatial_values^2)
s_value = 0.25 # s_value := k * ( (dt) / (dx)^2 )
# Uniform mesh
x_values = np.linspace(-2, 2, num_spatial_values)
# Time discretization
time_values = np.linspace(0, terminal_time, num_time_values)
# Definition of the solution u(x, t) to u_t = k * u_xx
u_values = np.zeros((num_spatial_values, num_time_values + 1)) # array to store values of the solution
# Finite Difference Scheme:
u_values[:, 0] = np.sin(np.pi * x_values) # x_values * (x_values - 1) # initial condition
for m in range(num_time_values):
for j in range(1, num_spatial_values - 1):
if j == 1 or j == num_spatial_values - 1:
u_values[j, m] = 0 ## Boundary condition
else:
u_values[j, m + 1] = u_values[j, m] + s_value * (
u_values[j + 1, m] - 2 * u_values[j, m] + u_values[j - 1, m]) ##Inner domain values
# Plot 3D surface
T, X = np.meshgrid(time_values, x_values)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_xlim3d(left=-2, right=2)
ax.set_ylim3d(bottom=0, top=1)
ax.set_zlim3d(bottom=-1, top=1)
surf = ax.plot_surface(X, T, u_values[:, :1600], rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=0, antialiased=False)
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()
3.6.4 Result¶
For the training set case with 50000 iteration, we easily note that the graph shape looks more similar to the actual one, than the case with 20000 iteration. The PINN (Physics-Informed Neural Network) method has the potential to improve the error bound as much as possible by increasing the number of iterations.
4. Reference¶
Files of tutorial :
[1] https://github.com/nanditadoloi/PINN/blob/main/solve_PDE_NN.ipynb
[2] https://services.math.duke.edu/education/baltimore/sample.html
[3] 'TwoDNSCH.py' file by Kevin
Paper :
[1] 'Proposal for quantitative benchmark computations of bubble dynamics' by S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman, S. Ganesank, and L. Tobiska
+) [2] 'Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks' by Colby L. Wight and Jia Zhao
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